Gluckstern, R L

Li, R

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Particle Accelerators, 1990, Vol. 29, pp. 159-165Reprints available directly from the publisherPhotocopying permitted by license only© 1990 Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaANALYSIS OF COAXIAL WIRE MEASUREMENT OF LONGITUDINAL COUPLINGIMPEDANCE1R.L. GLUCKSTERN AND R. LIPhysics Department, University of Maryland, College Park, MD, USAI. INTRODUCTIONIn recent work2 , a method has been developed to calculate the longitudinalimpedance for an azimuthally symmetric obstacle in a beam pipe of circularcross section. The method has been applied to a small obstacle3 , to anobstacle of general shape at high frequency2, and to several obstacles, 1n-clcding a periodic structure4,s at high frequency.Coupling impedances are difficult to measure directly. Instead, thereflection and transmission coefficients for a pulse carried through theobstacle on a thin coaxial wire are measured and the results apprOXimate thelongitudinal coupling impedance. 6In the present paper, an analysis is carried out including the coaxialwire and new boundary conditions for the fields at the surface of the wire.We then estimate the validity of the coaxial wire measurement for a varietyof frequencies and geometries. Finally, several numerical calculations arecarried out for both the bea~ and the wire pUlse. The results confirm thepredictions of the analysis.II. ANALYSIS FOR A BEAMLet us consider a beam pipe of radius a which enters and leaves an azi-muthally symmetric cavity of general shape. The longitudinal impedance canbe obtained by field matching at r = a. The source fields in the ultrarela-tivistic limit areZ Io 0 -Jkz- -- e2nr (2.1)Adding the pipe fields, we findJo(Kr)JOl) -jqz _Ez = dq A(q) e J (Ka) ,-00 0[861]/159(2.2)160/[862] R. L. GLUCKSTERN AND R. LI(2.3)Here k = ~/c, the suppressed time dependence is exp(j~t), Zo = 120n ohms, and10is the driving current. We have defined K = Jk2 - q2 and take the contourin the q plane below the poles on the negative real axis and above the poleson the positive real axis so that we have only outgoing waves for the pipefields as z ~ tm. Defining Ez(a,z) =fez), we havefez) = Jm dq A(q) e-j qz • A(q) = 2: J:dZ fez) e j qz •-m 0(2.4)where fez) vanishes for z < 0, z > g.The fields in the annular cavity region for r ~ a are expanded into anorthonormal set of cavity modes3 which satisfy metallic boundary conditionson the outer wall of the cavity as well 'as at r = a. Matching the magneticfield in the pipe and cavity regions leads towhere the pipe and cavity kernels are-jkzj e , (2.5)(2.6)Here ht(Z) is the normalized magnetic field in mode t at r = a,fez) = F(z) Z I /ka2, and J(q) is defined in Eq. (2.7). The sum in Eq. (2.6)o 0is over all azimuthally symmetric modes in the annular cavity.We can obtain a more explicit form for K (u) in Eq. (2.6) by using thepidentityJ'(Ka) 1A 0 00J(q)- Ka J (Ka) - 2 r 2 2 2 · (2.7)0 s=l q a bswhere Jsare the zeroes of Jo(x), and where b~ k2a2 _. J~ = -~~. For posi-tive u, the contour in Eq. (2.6) can be closed in the lower half plane, en-LONGITUDINAL COUPLING IMPEDANCE [863]/161closing the poles at qa = bsand qa = -j~s.encloses the poles at qa = -bsand qa = j~s.For negative u, the contourThe result for K (u) is thenp2njK (u)p a Ls=l b s(2.8)where bs 7 -J~s when J s > ka.The longitudinal impedance of the cavity can be written asZ(k) 1 fex»·k 2n: Af k) 1 l lk---- = ---- dz e J z E (0 z) = Z I = ---2 dz F(z) e J Z .Zo ZoI o -CX) z' 0 0 ka 0(2.9)The solution of Eq. (2.5) for F(z') can then be used to obtain the impedanceby means of Eq. (2.9).III. ANALYSIS FOR ~ COAXIAL WIREWe now start with a TEM mode in the beam pipe including a coaxial wire ofradius ro'described byZ Io 0 -jkz-- e -E2n:r r (3.1)wh~re ~ stands for the coaxial wire case and where we have normalized to makeEq. (3.1) identical to Eq. (2.1). Adding the pipe fields, we findF (Kr)fex» ~ -jqz 0Ez = dq A(q)e F (Ka) ,-ex» 0Z I Ft(Kr)_0 0 e-jkz + jk fex» dq A(q) e- j qz -------2n:r K F (Ka) ,-00 0(3.2)(3.3)where the linear combinationsF (x )o(3.4)with ~ = Yo(Kro)/Jo(Kro)' are chosen to satisfy the boundary condition on thesurface of the wire. The contour in the q plane is as before in order thatwe only have outgoing waves for the pipe fields.162/[864] R. L. GLUCKSTERN AND R. LIThe analysis continues as in Section II. The only difference is thatJ(q) in Eq. (2.7) is now(3.5)The function J(q) is analytic in q or K. except for a second order pole atK=O (first order poles at q=±k). and first order poles at the zeros of thedenominator of Eq. (3.5). We can then write J(q) as a sum over these polesby finding the appropriate residues. obtaining finally<Xco S- 2 L 2 2 -2's=O q a - bs1(3.6)(3.7)Here is is the value of Ka at the zeroes of Fo(Ka). with io- o. The pipekernel is thereforeThe expression for the magnetic field in the cavity region is identicalto that for the case of a beam on axis. since the boundary conditions are notaffected by the presence of the wire. The integral equation for the axialelectric field at the beam pipe is therefore-jkzj e • (3.9)with K being given by Eq. (3.8) and K by Eq. (2.6).P cIn order to obtain the transmission and reflection coefficients. it issimplest to examine Eq. (3.3) for large positive and negative z. The TEMLONGITUDINAL COUPLING IMPEDANCE [865]/163(coaxial) modes correspond to the poles at q = ±k. Specifically one obtainsZ lITEMo </>Z Io 02nr +2n a A(±k) e+ j kzor(3.10)where the ± 1s for the pipe region with z ~ O. Using Eq. (2.4), we obtainfor the reflection and transmission coefficients2na1 - T(k) = ---i !:dZ F(z) e j kz •ka 02naR(k) = ---i !:dZ F(z) e-j kz •ka 0(3.11 )where we have again used fez) F(z) Z I /ka2.o 0Comparison of Eq. (3.11) with Eq. (2.9) shows why the impedance corres-ponds more closely to the transmission coefficient rather than the reflectioncoefficient, particularly at frequencies for which kg ~ 1. In fact the cor-respondence isn1 - T(k) ~ In(a/r )oZ(k)Zo(3.12)(4.1)IV. COMPARISON OF I-T(k) AND 2naoZ(k)/ZoThe difference between the coupling impedance and 1 - T(k) for the pulse onthe wire is totally contained in the modified pipe kernel in Eq. (3.8).Specifically we have an additional term proportional to [In(a/r )]-1, a shiftoof the zeroes from Jsto is ' and the modified coefficients as' Forro/a « 1, it is easy to show thatis ~ js + R/2Ls ' s ~ 1 , Ls = in(2a/rojs) - 7 ,-1 2where 7 = .577 is Euler's constant. Also a ~ 1 - n/2J L for s ~ 1. Thus-r s sall changes are proportional to lin{a/ro)] or smaller, suggesting thatthere may be differences of order 20%, even for ro/a as small as .01.The result for a small obstacle of cross sectional area a may be takendirectly from earlier work. 3 Specifically we can write-jb g/a2n sa J a e ztn2]02nka [-CX) s1 - T(k) k 211+ E + J -n- (4.2)s=O bs164/[866] R. L. GLUCKSTERN AND R. LIWe therefore expect that the primary difference for a small obstacle will bea shift in the frequency at which the singular behavior occurs from ka = jsto is' This is confirmed in Figs. la and Ib where we pl9t the real andimaginary parts of Y(k) and 2n«o[1 - T(k)]-1/2ofor a pill box of lengthg = .0Sa and width b-a = .1a for ro/a = .01. The corresponding results forro/a = .1 are shown in Figs. 2a and 2b. The numerical results are obtainedwith programs which expand the fields in the pipe region and the cavity pluspipe region into traveling axial waves. The figures clearly show that thedetails of the two results differ, but that the average over the sawtoothbehavior is essentially unmodified. 7The result at high frequency is similarly easy to predict. In earlierwork, we showed2 that the average behavior of the impedance at high frequencyis obtained by converting the sum over s in the pipe kernel to an integral2 1/2over s, and that the main contributions come from s of order (ka /g) .Since the spacing of the zeroes is essentially unmodified, we expect no sig-nificant difference in the average behavior at high frequency. This is con-firmed in Figs. 3a and 3b for the real and imaginary parts of Z(k) and20[1 - T(k»)/2n«o for g/a = n/4, b/a = 1.5, ro/a = .1. What is remarkableis that the complicated oscillatory behavior is duplicated as well.V. SUMMARYWe have derived the integral equation for the transmission coefficient of thecoaxial mode for a pulse along a wire on the axis of a beam pipe and cavity.The only difference between this equation and the one for the longitudinal-1coupling impedance is in the pipe kernel and is of order linea/roll orless. Specific predictions are made for the comparison between Z(k)/Zo and[1 - T(k)]/2n«o for both a small obstacle, and for a larger obstacle at highfrequency, and these are confirmed by numerical calculations. Our conclusionis that measurement of 20[1 - T(k)]/2n«o corresponds remarkably well to theactual longitudinal coupling impedance.VI. ACKNOWLEDGMENTThe authors would like to express their appreciation to Filippo Neri forhelpful conversations.VI I. REFERENCES1. Work supported by the Department of Energy.2. R.L. Gluckstern, Phys. Rev. D, 39, 2732 (1989).3. R.L. Gluckstern and F. Neri, Longitudinal Coupling Impedance of a SmallObstacle, Proceedings of the Particle Accelerator Conference, Chicago,LONGITUDINAL COUPLING IMPEDANCE [867]/165IL, March 1989.4. R.L. Gluckstern, High Frequency Behavior of the Coupling Impedance for aLarge Number of Obstacles, Proceedings of the Particle Accelerator Con-ference, Chicago, IL. March 1989.5. R.L. Gluckstern, Phys. Rev. D, 39. 2780 (1989).6. See, for example, H. Hahn and F. Pederson, On Coaxial Wire Measurementsof the Longitudinal Coupling Impedance. BNL Report No. 50870 (1978).7. The erratic nature of a few of the singularities is due to the cut off inmode number and the frequency step used in the numerical calculation.0.80.200.. "19 • .OSa- ::.b c: 1.1a· · : '0 •.01.1-I ~ i··· __ ··--·--4··········· ..~ -" .. ".1,•• • . • . . .::' 1 1-15 ••..·,··.-.;-.••••••..••r-..- t.-- -.. -·.t .0.0 -+-------.........---+--.--+----+-------i -2 I , --.-,--+------t-'----1o 5 10 ka 15 20 25 0 5 10 ka 15 20 25Figure 1a. Real part of admittanc~ for a beam (solid (urv~) Figure 1b. Imaginary part of admittance for a beam (solid curv~)and a coalial wir~ of radius,o =.O'a (dashed curve). and a coaxial wire of radius ro :: .01a (dashed curve).0.10.2O!'l , t .. .t ········r········~;·:~ 1. :~········1········--··f-·······-·[f:·~~:-.1 _. .! i--._ _.. 1- ··.L.... .. .. i~~.~.:~· .~: :.~.:_~J:.~=t- .. -.-.-.---r~-.~~.~j:_~-~._-~0.0 -I-0--~·-+6----1t-0--Ie-.--I15t------120t-----i2S 0 5 10 ka If) 20 2!»Figure 2a. Real part of admittance fOf a beam (solid curve) Figure 2b. Imaginary part of admittance for a beam (solid curve)and a coaxial wire of radius'o •.1a (dashed curve). and a coaxial wire of radius '0 c .1a (dashed curve).9 • n.,4b.l.5a'0•.1.-15 ..f----+----+-----I-------1~---i40 20 2" 28 Ie. 32 36 40Figure 3b. Imaginary part of impedanc~ for a beam (sohdcurve)and coaxial wire of radius '0 •. 1a (dashed curve).. ',o ------t----i-....-----.jt-----1--2() ~4 ~It Ita 32 36Figure 3a. Rea' part of impedance for a beam (solid cutve)and coali., wire of radius ro. ,. (dashed curve)eo f)g• "AI •• '.Sae to •.1.&. 00:;;N e"It:.&;010· :vN!-105-

Analysis Of Coaxial Wire Measurement Of Longitudinal Coupling impedance

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