MModeling same-order modes of multicell cavities
Olof Troeng ∗ Department of Automatic Control, Lund University, Sweden (Dated: February 11, 2021)We derive the transfer function of a multicell cavity with parasitic same-order modes (from powercoupler to pickup probe). The derived model is discussed and compared to measurement data.
I. INTRODUCTION
Multicell elliptical cavities are suitable for acceleratingparticle beams with velocities greater than 0 . c . However,these cavities intrinsically have parasitic electromagneticmodes that are close in frequency to the acceleratingmode. These so-called same-order modes , or fundamentalpassband modes, interact both with the rf system and thebeam. Their interaction with the rf system may, withoutcounter-measures, give instability in the field control loop[1], and their interaction with the beam drives emittancegrowth [2]. Dynamic models of same-order modes andthese interactions are necessary for design and analysis offield control algorithms.The regular geometry of multicell elliptical cavitiesmakes it possible to derive the same-order-mode dynamicsfrom a small number of cavity parameters. Such dynamicmodels have been used for studying how beating of same-order modes affects the beam [3]. The steady-state modelsin [4–7] capture the shapes and resonance frequencies ofthe modes but not their dynamics and decay rates.For field control analysis in the frequency domain, itis necessary to know the transfer function from the rfdrive to the pickup-probe signal. In [1], a real-coefficient,two-input two-output transfer function from rf drive topickup signal (valid for superconducting cavities) waspresented without motivation. Such real-coefficient, two-input two-output models are common in the field controlliterature. However, as we discussed in [8], the equivalent,complex-coefficient, single-input single-output (SISO) rep-resentation gives more intuition and simplifies analysis.In this paper we: (1) derive a complex baseband modelof a multicell cavity along the lines of [3], but using theenergy-based parameterization from [9]; (2) normalizethat model to make it suitable for field control analysis;(3) derive the complex-coefficient SISO transfer functionfrom power coupler to pickup signal (also valid for normal-conducting cavities); and (4) fit the transfer-functionmodel to measurement data from a 6-cell niobium cavitytaken at room temperature and cryogenic temperature. Remark 1:
An incorrect transfer-function model for cav-ities with parasitic modes was proposed in [10]. It wasbased on taking the transfer function of the individualmodes from [1] and multiplying them with the modes’coupling strengths to the rf system, although that effect ∗ E-mail: [email protected]
Rf drive: F g Pickup signal: V pu Beam current: I b Same-order modes: A π (shown), A π/ , . . . A π/ FIG. 1. Schematic of 6-cell elliptical cavity. Power is fedthrough the power coupler in the leftmost cell and the cavityfield is sensed by a pickup probe in the rightmost cell. was already accounted for in [1]. The incorrect modelin [10] predicts that the resonance peaks have differentmagnitudes, while both the analysis in this paper and[11][Fig. 5.9] indicate that they have similar magnitudes.Also the stability analysis in [10] is problematic. Itis not possible to analyze closed-loop stability of MIMOsystems with strong cross couplings (as from same-ordermodes) using loop-by-loop analysis [12, Sec. 8.6], [13].The complex-signal perspective in [8] together with themodel in this paper enables correct and intuitive analysis.
II. SAME-ORDER MODES OF MULTICELLCAVITIESA. Cavity model with parasitic modes
We start by considering parasitic modes of a generalcavity using the energy-based parameterization in [9, 14].Let A k be the complex envelope of mode k , with its mag-nitude (units √ J) scaled so that | A k | is the mode energy.This unambiguously defines the mode amplitude withoutreference to the effective voltage of the accelerating mode.If coupling between the modes can be neglected then theircomplex envelopes evolve according to [9] d A k dt = ( − γ k + i ∆ ω k ) A k + (cid:112) γ ext k F g + α k I b , (1a)where γ k = γ k + γ ext k is the decay rate of the modeamplitudes ( γ k corresponds to resistive losses and γ ext k to decay through the power coupler); ∆ ω k = ω k − ω rf is the offset of the mode frequency ω k relative to thenominal rf frequency ω rf ; F g (units √ W) is the envelopeof the forward wave from the rf amplifier with | F g | equalto the power in the wave; and I b is the beam phasor, with | I b | equal to the dc beam current. The complex envelopes a r X i v : . [ phy s i c s . acc - ph ] F e b A k are defined relative to ω rf , with their phases chosen sothat the coefficients in front of F g are real. The voltagesensed by the pickup probe is a linear combination V pu = N (cid:88) k = a, , ,... C k A k (1b)of the mode amplitudes, where C k are complex coefficients.Here we have used the subscript a to indicate the modeintended for acceleration. We will use a different labelingwhen discussing same-order modes.The cavity–beam-coupling parameters α k are in gen-eral complex. However, the parameter α a = α a for theaccelerating mode is real due to the definition of I b . Normalized model with parasitic modes
In many situations, e.g., field control analysis, it iseasier to work with normalized models. Therefore, we in-troduce the following dimension-free, normalized variablessimilarly to in [9], a k := 1 A a A k (2a) f g := 1 γ a A a (cid:112) γ ext a F g (2b) i b := 1 γ a A a α a I b (2c) v pu := 1 C a A a V pu , (2d)where A a is the nominal magnitude of the acceleratingmode. With the variables (2) we can write (1) as d a k dt = ( − γ k + i ∆ ω k ) a k + γ a √ γ ext k √ γ ext a f g + γ a α k α a i b . (3a) v pu = a a + N (cid:88) k =1 , ,... c k a k (3b)where c k := C k / ( C a A a ). The normalization (3) has thenominal operating point a a = 1 and the steady-statesensitivity of a a to variations in f g and i b is unity. B. Same-order modes of multicell elliptical cavities
The same-order modes of a multicell cavity arise fromthe coupling between the cells’ fundamental modes, sim-ilarly as for a chain of weakly coupled oscillators. An N -cell cavity has N closely spaced same-order modes. Theparameters in (3) of these modes, can, due to the regularcavity geometry, be computed from a small number ofbasic cavity parameters, as shown in the Appendix. Themodes are conventionally referred to as the π/N mode, TABLE I. Values of R n for different n and N , see (4) for R n . N n the 2 π/N mode, up to the π mode. This naming indicatesthe cell-to-cell phase advance of the sinusoidal envelopeof the mode shapes . It is typically the π mode that isused as the accelerating mode [5]. We will use a subscript π to indicate parameters of the π mode and a subscript n for the nπ/N mode.The derivation in the appendix assumes an “ideal” N -cell cavity in that: all cell-to-cell coupling factors equal k cc ; all inner-cells have resonance frequencies equal to ω cell ; the end-cell resonance frequencies are tuned for aflat π mode; and mode coupling is negligible. Under theseassumptions, the same-order-mode parameters depend, inaddition to N , ω cell , and k cc , only on the cell’s resistivedecay rate γ and the decay rate γ pc through the powercoupler from the connected end cell. By introducing R n := √ nπ N if n < N n = N , (4)the parameters can be expressed as∆ ω n = ω cell (cid:112) R n k cc − ω rf ( n < N ) (5) ≈ ω cell ( (cid:112) R n k cc − (cid:112) k cc ) (5 (cid:48) ) γ n = γ (6) γ ext n = R n γ pc /N = R n γ ext π (7) c n = ( − N − n R n . (8)The approximation (5 (cid:48) ) assumes that ∆ ω π = ω π − ω rf is small relative to ∆ ω N − , see also Remark 2. Withthe relationships (5)–(8) the general model in (3) takesthe form in Fig. 2. Values of R n for different valuesof N and n are shown in Table I. In the next sectionwe investigate how well the relations (5)–(8) agree forparameters estimated from a real-world cavity. Remark 2:
While ∆ ω π is negligible relative to both∆ ω N − and typical field control bandwidths, its precisetuning, typically to a value slightly larger than 0, is crucialfor minimizing the drive power | F g | [5, 6, 9]. Remark 3:
Typical cell-to-cell coupling factors k cc are The cell-to-cell phase difference of the modes themselves is 0 or π . γ π s + γ π − i ∆ ω π f g i b γ π s + γ N - − i ∆ ω N - α N - α π R N - − R N - a N - γ π s + γ − i ∆ ω α α π R (-1) N - R a γ π s + γ − i ∆ ω a π ...... ... ... v pu FIG. 2. Block diagram for a normalized model of a cavitywith parasitic same-order modes. Subscripts n indicate the nπ/N mode. on the order of 0 .
01, which gives∆ ω n ≈ ω cell (cid:16)(cid:112) R n k cc − (cid:112) k cc (cid:17) ≈ ( R n − k cc ω cell . This shows that the baseband frequencies of the same-order modes are approximately proportional to R n − Remark 4:
There are two definitions of the cell-to-cellcoupling factor k cc in the accelerator literature. The full-passband-width definition, for which k cc ≈ ( ω π − ω ) /ω π [4, 6, 7, 11] and the half-passband-width definition [3, 5] forwhich k cc is half as large. The first definition correspondsto the per-cycle decay of a cell’s energy due to couplingto a neighboring cell and the latter definition correspondsto the decay of the field amplitude. In this paper we usethe half-passband-width definition since it gives slightlymore convenient expressions. C. Transfer function from rf drive to pickup probe
The transfer function from f g to v pu in Fig. 2 is impor-tant for field control analysis. It is given by P cav ( s ) = γ π N (cid:88) n =1 ( − N − n R n s + γ n − i ∆ ω n = γ π γ ext π N (cid:88) n =1 ( − N − n γ ext n s + γ n − i ∆ ω n , (9)where γ n := γ + γ ext n is the total decay rate of the nπ/N mode. For superconducting cavities with γ (cid:28) γ ext n we have that γ ext n ≈ γ n and (9) simplifies to P cav ( s ) = N (cid:88) n =1 ( − N − n γ n s + γ n − i ∆ ω n . (10)Let us observe some characteristics of the transfer func-tion (9). First, since the numbers γ n are small relative tothe differences between the numbers ∆ ω n , the transferfunction P cav ( s ) has sharp resonance peaks at (baseband)frequencies ∆ ω n . Also note that ∆ ω n < n < N ,i.e., the baseband resonance frequencies of all parasiticsame-order modes are negative.For superconducting cavities ( γ (cid:28) γ ext n ) we see from(10) that all peaks have approximately unity magni-tude. For normal conducting cavities ( γ (cid:29) γ ext n ) wehave that the peak magnitude of the nπ/N mode equals γ ext n /γ ext π = R n (see Table I). Fig. 4 shows the Bodemagnitude plots of these extreme cases. III. COMPARISON TO MEASUREMENT DATA
In this section we examine the fit of relationships (5)–(9)to network-analyzer measurements of a 6-cell medium- β cavity for the European Spallation Source (ESS). Mea-surements were taken when the cavity was normal con-ducting (NC, at room temperature) and superconducting(SC, at 2 K). In the first case γ (cid:29) γ ext n and in the lattercase γ ≈
0. The measurements (scaled and frequencyshifted for a unity-gain π mode at zero frequency) areshown in gray in Fig. 3.The observed same-order-mode frequencies (relativeto the π mode) are shown on the frequency axes inFig. 3. From those values, the cell-to-cell coupling fac-tor (half-passband-width definition) was estimated to k cc ≈ . ω n in (5)agree with the observed frequency offsets to within ± ω n , the remaining parameters in (5)–(8), namely γ and γ ext π , were estimated by fitting (9) to the data.A good fit was obtained with γ ext π / π = 460 Hz, γ = 0(SC), and γ / π = 35 kHz (NC). Figure 3 shows the fittedmodels and the measurement data.For comparison, we estimated γ k and the peak magni-tude g k for each individual mode by fitting g k γ k / ( i ( ω − ω k ) + γ k ) to the data in the vicinity of each mode. InTable II, these “observed” mode parameters are comparedto those predicted by (6), (7), and (9). It is seen thatthey agree reasonably well—in particular for the modesclosest to the π mode, which are the most crucial ones The cavity was a prototype without a tuning system, hencethe mode frequencies during the measurements differed fromthe design frequencies. For the π -mode, which should have anominal frequency of 704 .
42 MHz, the measured frequencies were703 .
26 MHz (NC) and 704 .
24 MHz (SC). That the resonance fre-quency is significantly lower at room temperature is typical. − . ω − . ω − . ω − . ω − . ω ω π − − Baseband Frequency ω/ π (MHz) | P cav ( iω ) | (a) Superconducting (2 K). − . − . − . − . − .
784 010 − − Baseband Frequency ω/ π (MHz) | P cav ( iω ) | (b) Normal conducting (room temperature).
FIG. 3. Network-analyzer measurements of the transmissionof a 6-cell ESS medium- β cavity (gray) and fits of the transferfunction (9) (blue, orange). The data is scaled for unity gainof the π mode. Measurements by P. Pierini, ESS.TABLE II. Comparison between peak magnitudes and band-widths for measurement data and model fits in Fig. 3. Theobserved values are shown and the value in parenthesis indi-cates the deviation from the value predicted by the model.Superconudcting Normal conducting | P cav ( i ∆ ω k ) | γ k / π | P cav ( i ∆ ω k ) | γ k / π (Hz) (kHz) π (1 . ( − π/ . ( − ( − . ( − . (3%) π/ . ( − ( − . (0 . (3%) π/ . ( − ( − . ( − (3%) π/ . ( − ( − . ( − (5%) π/ . (7%) ( − . ( − (9%) in a field control context. The small discrepancies areprobably explained by variations in the cell parameters.For field control analysis, it is convenient to plot fre-quency responses using a logarithmic frequency axis, inso-called Bode diagrams. The frequency response of a nor-mal conducting and a superconducting cavity are shownin a double-sided Bode diagram in Fig. 4. Note that su-perconducting cavities, whose external decay rates γ ext π are significantly larger than γ , have resonance peaks withapproximately unity magnitude (Fig. 4). For normal con- − − − | P cav ( iω ) | NCSCBaseband Frequency, ω/ π (Hz)FIG. 4. Bode magnitude plots of the transfer function (9)for a 6-cell cavity when it is superconducting (SC, γ = 0)and normal conducting (NC, γ / π = 35 kHz). Note that thetransfer functions have been scaled for unity magnitude at thezero frequency, see Remark 5. ducting cavities (with small γ ext π ), the peak magnitudesare approximately given by R n . Remark 5:
Recall that the data in Fig. 3 and Fig. 4 isscaled for unity magnitude of the π mode. In absoluteterms, the transmission of the normal conducting cavityis lower than for the superconducting cavity. IV. CONCLUSION
We have derived a complex-coefficient transfer functionmodel that is valid for both normal conducting and super-conducting cavities. The model was seen to give a goodfit to measurement data.
ACKNOWLEDGMENTS
Bo Bernhardsson and Paolo Pierini contriubted helpfulcomments and suggestions. The measurement data inSec. III was provided by Paolo Pierini. The author is amember of the ELLIIT Strategic Research Area at LundUniversity.
Appendix: Derivation
In this appendix, we will start from the bandpass state-space model of an N -cell cavity in [3], perform modaldecomposition (diagonalization), and then transform thediagonal model to baseband.
1. Bandpass model of an N -cell cavity Our starting point is the standard model for studyingsame-order modes of multicell cavities [3, 5, 7], but wewill use slightly different notation. Consider the elliptical N -cell cavity in Fig. 1 which has N − N is sensed by a pickup probe mountedin the beam pipe.We only consider the lowest-energy mode in each celland we denote the electric field amplitudes of these by x = (cid:2) x · · · x N (cid:3) T . We assume that x (cid:96) is normalized so thatthe squared magnitude of its complex envelope equals theenergy stored in cell (cid:96) . Let all cell-to-cell coupling factorsbe given by k cc ; the inner-cell resonance frequencies begiven by ω cell ; the end-cell resonance frequencies be givenby ω cell √ k cc ; the rf drive (i.e., the forward waveentering the power coupler) be modeled by its complexenvelope F g , with | F g | equaling the power in the forwardwave; the coupling between the waveguide and the fieldin cell 1 be quantified by the decay rate γ pc of the fieldin this through the power coupler; and assume that fielddecay through the pickup probe is negligible. Accordingto [3, (B-1)] the field amplitudes x in the cells evolve asa chain of weakly coupled oscillators, with the dynamics¨ x + 2 γ ˙ x + 2 γ pc E ˙ x + ω x + ω k cc Kx = 2 (cid:112) γ pc e ddt Re { F g e iω rf t } , (A.1)where K = − · · · − − − − · · · − , (A.2) E = diag(1 , , . . . , ,e = (cid:2) . . . (cid:3) T .
2. Eigenvectors and eigenvalues of the matrix K By recalling standard trigonometric identities and doingsome algebra—alternatively looking up [5, Sec. 7.2]—it can be verified that the matrix K in (A.2) has theeigenfactorization Q Λ Q T = K , where Q := | | | q q . . . q N | | | , (A.3)Λ := diag( λ , λ , · · · , λ N ) , (A.4)and the eigenvalues λ n are given by λ n = 2 (cid:16) − cos nπN (cid:17) , (A.5) − . . π/ n = 1) − . . π/ − . . π/ − . . π/ − . . π/ − . . π mode ( n = 6)FIG. 5. Same-order-mode shapes q n (A.6) of a 6-cell cavity. and the orthonormal eigenvectors q n are given by q n = (cid:114) N ( n < N ) sin (cid:2) (1 − ) nπN (cid:3) sin (cid:2) (2 − ) nπN (cid:3) ...sin (cid:2) ( N − ) nπN (cid:3) , q N = (cid:114) N − − N − . (A.6)The the mode shapes (A.6) are illustrated in Fig. 5. Asmentioned in Sec. II B, mode n is often referred to as the nπ/N mode. The entries of the N th mode have equalmagnitude and opposite signs and for this reason it isalmost always used as the accelerating mode.
3. Diagonalizing the dynamics
Letting ξ be the passband mode amplitudes ( x = Qξ ),we may diagonalize all of (A.1) except the third term,¨ ξ + 2 γ ˙ ξ + 2 Q T E Qγ pc ˙ ξ + ω ( I + k cc Λ) ξ = 2 (cid:112) γ pc Qe ddt Re { F g e iω rf t } , (A.7)where I is the identify matrix. For convenience, denote Q ’s first row times √ N by R := (cid:2) R · · · R n · · · R N (cid:3) = (cid:104) √ π N · · · √ nπ N · · · (cid:105) , that is R n is given by (4). Then we can write the thirdterm of (A.7) as 2 R T R/N γ pc ˙ ξ . This term, which origi-nates from field decay through the power coupler, dynam-ically couples the modes. We may however assume thatthis interaction averages out to 0, since the beating periodbetween the different modes is significantly shorter thanthe timescales at which the (complex) mode amplitudeschange. Thus it suffices to keep the diagonal entries of R T R , which correspond to the external decay rates of themodes. Denoting them by γ ext n := R n γ pc /N, (A.8)we get N uncoupled differential equations from (A.7), onefor each mode,¨ ξ n + 2 γ ˙ ξ n + 2 γ ext n ˙ ξ n + ω (1 + k cc λ n ) ξ n = 2 (cid:112) γ ext n ddt Re { F g e iω rf t } . (A.9)From (A.8) we see that the external decay rate of the π mode is given by γ ext π = γ pc /N and hence the externaldecay rates of the same-order modes satisfy γ ext n = R n γ ext π . (A.10)
4. Baseband state-space model
The eigenfrequencies of the modes in (A.9) are ω n = ω cell (cid:112) k cc λ n . See Fig. 7.4 in [5] for an illustration. Denote their offsetsfrom the nominal rf frequency ω rf by ∆ ω n := ω n − ω rf .Let A n denote the complex envelope of the nπ/N modewith respect to ω rf , i.e., ξ n = Re { A n e iω rf t } . A slowly-varying envelope approximation of (A.9) is then givenby˙ A n = [ − ( γ + γ ext n ) + i ∆ ω n ] A n + (cid:112) γ ext n F g . (A.11)By introducing A := (cid:2) A · · · A N (cid:3) T , ∆Ω :=diag(∆ ω , . . . , ∆ ω n ), and Γ ext := diag( γ ext1 , . . . , γ ext N ), we can write the equations (A.11) as˙ A = [ − ( γ I + Γ ext ) + i ∆Ω] A + (cid:112) γ ext π R T F g . (A.12)The voltage signal V pu from the pickup probe isproportional to the field amplitude in cell N , i.e., V pu ∝ Q N : A where Q N : denotes the N th row of Q . Us-ing that √ N Q Nn = ( − n − R n (which follows from basictrigonometry), we can write V pu = C A , where C = (cid:2) c · · · c N - c π (cid:3) := κ pu R ( − N − ... − , (A.13)and κ pu is a proportionality constant which can be as-sumed to be real (since the reference phase of V pu canbe chosen freely).Combining (A.12) with (A.13) and also including cavity–beam interaction, quantified by parameters α n as in Sec-tion II, we have the following state-space realization ofthe same-order-mode dynamics˙ A = A A + B g F g + B b I b (A.14a) V pu = C A (A.14b)where A = − ( γ I + Γ ext ) + i ∆Ω (A.14c) B g = (cid:112) γ ext π R T (A.14d) B b = 12 (cid:2) α · · · α N -1 α π (cid:3) T (A.14e) C = given by (A.13) . (A.14f) [1] T. Schilcher, Vector Sum Control of Pulsed AcceleratingFields in Lorentz Force Detuned Superconducting Cavities ,Ph.D. thesis, University of Hamburg, Germany (1998).[2] R. Ainsworth and S. Molloy, Studies of parasitic cav-ity modes for proposed ESS linac lattices, in
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