aa r X i v : . [ phy s i c s . acc - ph ] S e p Energy-based parameterization of accelerating-mode dynamics
Olof Troeng ∗ Department of Automatic Control, Lund University, Sweden (Dated: October 1, 2020)We propose a parameterization of the accelerating-mode dynamics of accelerating cavities in termsof energy, power, and current. This parameterization avoids many confusing features of the popularequivalent-circuit-based parameterization where a fictitious generator current is introduced. Tofurther simplify analysis and understanding of the accelerating-mode dynamics we also propose aconvenient normalization and use phasor diagrams for illustrations.
I. INTRODUCTION
Radio-frequency (rf) particle accelerators acceleratebunches of charged particles using oscillating electromag-netic fields that are confined in rf cavities. Analysisof field transients and field control algorithms require amodel of the field dynamics. Rf cavities have infinitelymany electromagnetic eigenmodes, but in many situa-tions it is sufficient to model the so-called acceleratingmode intended for particle acceleration.Since the accelerating mode provides a voltage to thecharged particles, and the particle beam corresponds toa current, it is natural to model the accelerating modeas an equivalent (electric) circuit. This is the standardapproach to model accelerator cavities [1, 2] and is alsocommon in rf engineering [5, 6].However, equivalent-circuit-based parameterizations ofthe accelerating mode have a number of inconvenient fea-tures [2]. These mainly stem from that the rf drive needsto be considered as a virtual current to fit it into theequivalent-circuit framework. A parameterization thatavoids these issues was proposed by Haus for modelingoptical cavities [7]. In Haus’ parameterization the ampli-tudes of the accelerating mode and the rf drive are takenas the square root of the mode energy and the squareroot of the forward power, respectively. These quantitiesare, unlike those in equivalent-circuit-based parameteri-zations, defined independently of the beam velocity andthe strength of the cavity–waveguide coupling.In this paper we extend Haus’ energy-based parame-terization [7] to include beam loading, which enables itto model accelerating cavities. We then discuss its ad-vantages over equivalent-circuit-based parameterizations.We also propose a normalization of the cavity dynam-ics that simplifies the analysis of field control algorithms. ∗ E-mail: [email protected] Much of the existing literature has an emphasis on steady-staterelations [3, 4], which are not helpful for field control analysis. For example, as was noted by T¨uckmantel [2], “for considera-tions where Q ext varies—as for a variable coupler e.g. in contextwith rf vector feedback loop gain—or where ( R/Q ) varies—aswhen particles of different speed v = βc pass the same cavity /.../the model currents cannot be considered constant; they have tobe re-adapted each time Q ext or ( R/Q ) change”.
Envelope of the acceleratingcavity mode: A ( V )Rf drive: F g ( I g ) (Forward wave) Reverse wave: R g Beam current: I b ( I b , rf )FIG. 1. Illustration of an accelerating rf cavity coupled to aa waveguide. The cavity field is established and maintainedby the forward wave F g that is provided by an rf amplifier( g enerator). The clearly colored letters denote the complexenvelopes that are used in the proposed energy-based param-eterization. Parenthesized gray letters indicate the variablesthat are typically used in equivalent-circuit-based parameter-izations. Throughout, we illustrate the discussed concepts with asomewhat novel type of phasor diagrams.
Notation and assumptions: (1) The considered system isshown in Fig. 1. We restrict our attention to the acceler-ating cavity mode and will not consider parasitic modes.(2) Rf signals are represented by their complex envelopes(equivalent baseband signals), which are denoted by boldletters. (3) Particle bunches are assumed to be point-like(it is straightforward to include a relative bunch formfactor [2]).
II. BACKGROUND: EQUIVALENT-CIRCUITBASED PARAMETERIZATIONS
Two popular, equivalent-circuit-based parameteriza-tions in the existing literature are: Tckmantel’s [2] , d V dt = − (cid:20) ω a Q ext + ω a Q − i ∆ ω (cid:21) V + ω a r ◦ Q I g + ω a r ◦ Q I b , rf , (1a) Eq. (53), with minor modifications for consistency. and Schilcher’s [1] , d V dt = ( − ω / + i ∆ ω ) V + R L ω / (2 I g + I b , rf ) . (1b)In the above equations, V denotes the complex enve-lope of the effective accelerating voltage of the cavity field; I g denotes the “generator current” which models the rfamplifier drive; and I b , rf denotes the rf component of thebeam current ( | I b , rf | = 2 I dc [1, A4]), see Tables Ia and Ibfor a complete list of the quantities involved. The normal-ized shunt impedance r ◦ /Q in (1a) is defined with respectto the equivalent-circuit convention , and so is the loadedresistance R L in (1b). There is also the linac convention for which r/Q = 2( r ◦ /Q ), [2].Note that the generator current I g is a fictitious quan-tity that is introduced to make the rf drive term fit intothe equivalent-circuit framework. An additional relationis needed for how I g relates to the drive power P g , whichis the physical quantity of interest, P g = 12 r ◦ Q Q ext | I g | . (1c) III. ENERGY-BASED PARAMETERIZATION
Let the state of the accelerating cavity mode be quanti-fied by the complex-valued mode amplitude A , with | A | equal to the stored mode energy ( A has units √ J). Themode amplitude A is related to the effective acceleratingvoltage via V = α A , where α = p ω a ( r/Q ) quantifies the coupling betweenthe cavity field and the beam. Recall that α , just like( r/Q ), depends on the beam velocity [4]. With thesedefinitions the dynamics of the accelerating mode can bewritten d A dt = ( − γ + i ∆ ω ) A + p γ ext F g + α I b , (2)where γ = γ + γ ext is the decay rate of the cavityfield ( γ corresponds to resistive losses and γ ext to de-cay through the power coupler); F g is the envelope ofthe forward wave from the rf amplifier with | F g | equalto the power in the wave ( F g has units √ W); and I b isthe beam-loading phasor, with | I b | equal to the dc beamcurrent. Remark 1:
If an equation in V is desired, multiplying(2) by α gives d V dt = ( − γ + i ∆ ω ) V + α p γ ext F g + α I b . (3)Most of the advantages mentioned in the next subsectionapply to this parameterization as well. Eq. (3.49) interpreted for complex signals, with I = 2 I g + I b , rf . Remark 2:
In the proposed parameterization (2) we con-sidered the factor γ as a decay rate rather than as abandwidth as in (1b). This is consistent with the laserliterature [8] and makes it natural to write the total de-cay rate as γ = γ + γ ext ; a relation that would be lessintuitive in terms of bandwidths. For frequency-domainconsiderations one should, of course, think of γ as a band-width. Remark 3:
The quality factor (Q factor) of an oscillatorquantifies its decay in terms of oscillation periods. Q fac-tors are popular [2, 3, 9, 10] for quantifying the decay ofrf cavity modes, e.g., Q , Q ext , and Q L in Table Ib. How-ever, in most situations where the field dynamics are ofinterest, such as field control, it is the absolute timescalesthat are of interest. In these situations Q factors providelittle information on their own (before division with ω rf ).The decay rates γ , γ ext , and γ in Table Ic, on the otherhand, capture all relevant information and are meaning-ful in their own right. Remark 4:
As indicated in Fig. 1, there is a reverse wavepresent in the waveguide. The complex envelope of thereverse wave (units √ W) is given by (see Appendix 8) R g = − F g + p γ ext A . (4)Under ideal steady-state conditions R g = 0, see Sec. V. Comparison of the energy-based parameterization (2)to equivalent-circuit-based parameterizations
With Table I it is easy to verify that the three param-eterizations (1a), (1b) and (2) are equivalent. Note thatalso field control requirements of the form x % amplitudeerror and y ◦ phase error are identical for V and A Below we give some pros and cons of the proposedparameterization (2) relative to the parameterizations(1a) and (1b).
Advantages of the proposed parameterization (2) are:1. The dynamic equation is cleaner. Compare for exam-ple the expression in (1c) to P g = | F g | . Remark:
Quantifying signal amplitudes in terms ofsquare root of power, as for F g , is common in rf engi-neering. It is with respect to such power waves thatscattering matrices are defined [6]. Example:
The rf drive power necessary to maintainan accelerating voltage V = α A , while acceleratinga beam modeled by I b0 , is easily found from (2) as P g = 12 γ ext (cid:12)(cid:12)(cid:12) ( − γ + i ∆ ω ) V /α + α I b0 (cid:12)(cid:12)(cid:12) . This expression is more convenient and easier to re-member than (1a/1b) together with (1c).
TABLE I. Physical quantities in the equivalent-circuit-basedparameterizations (1a)/(1b) and in the proposed parameter-ization (2). The rightmost column contains the quantitiesexpressed in the parameters of the other parameterization. a) Quantities common to (1a)/(1b) and (2) ω a rad / s Resonance frequency of the accelerating mode∆ ω rad / s Detuning of the accelerating mode, = ω a − ω rf b) Quantities in (1a)/(1b)V V Accelerating voltage V = α AI g A Generator current 2 √ γ ext /α F g I b , rf A Beam current (RF component) 2 I b Q – Unloaded quality factor ω a / (2 γ ) Q ext – External quality factor ω a / (2 γ ext ) β – Coupling factor, = Q /Q ext γ ext /γ Q L – Loaded quality factor, = Q / ( β + 1) ω a / (2 γ ) ω / rad / s Half bandwidth, = ω a / (2 Q L ) γr/Q Ω Normalized shunt impedance, α /ω a linac convention r ◦ /Q Ω Normalized shunt impedance, α / (2 ω a )equiv.-circuit convention, = ( r/Q ) / R L Ω Loaded shunt impedance, α / (4 γ )= ( r ◦ /Q ) · Q / (1 + β ) c) Quantities in (2)A √ J Mode amplitude V / p ω a ( r/Q ) F g √ W Rf drive p ( r/Q ) Q ext / I g I b A Beam current I b , rf / γ / s Resistive decay rate ω a / (2 Q ) γ ext / s External decay rate ω a / (2 Q ext ) γ / s Total decay rate, = γ + γ ext ω / (half bandwidth) α V / √ J Field–beam coupling parameter p ω a ( r/Q )
2. The impact of changes to the cavity–waveguide cou-pling γ ext and cavity–beam coupling α are transpar-ent. The same cannot be said for the parameteriza-tions (1a)/(1b). Example:
The quantity F g in (2) that represents therf drive is independent of γ ext . Contrast this to thedefinition of I g , implicitly given by (1c), that includesboth Q ext and r/Q . Thus, the parameterization (2)avoids the issue mentioned in footnote 2. Example:
From (1b) we see that the transfer functionfrom beam-current variations to cavity field errors is G I b → e ( s ) = ω / s + ω / R L . Recognizing the first factor as a low-pass filter withbandwidth ω / , one is led to believe that reducing ω / reduces the field errors from beam-current ripple.This is incorrect, however, since R L depends inverselyon ω / . This confusion does not arise from (2). 3. The mode amplitude A depends only the cavity field,while the effective cavity voltage V also depends onthe beam velocity [4]. Without a given beam velocity,the effective accelerating voltage V is not well-defined.For electron linacs, one could assume that the beamvelocity equals c and hence uniquely quantify the am-plitude of the accelerating mode by its effective volt-age. But the amplitudes of parasitic same-order modesof multi-cell cavities can obviously not be quantifiedthis way. For example, the same-order modes of theTESLA cavity all have a negligible coupling to thebeam, i.e., zero effective voltage, but they are crucialto consider in field-control analysis.4. The equation (2) can be derived using basic proper-ties of Maxwell’s equations (see the Appendix). Thisarguably allows for a better understanding of how themodel parameters relate to physical cavity properties. Disadvantages of the proposed parameterization (2) are:1. The parameterization (2) does not explicitly containthe accelerating voltage V , which is arguably the mostimportant quantity from a beam perspective. If it isnecessary with an equation in V one may use (3) whichretains many of the advantages of (2).Overall, there is less reason to use the energy-basedparameterization (2) when the focus is on beam sta-bility, as is often the case in circular machines. It isfrom an rf or field stability perspective that (2) bringshelpful intuition.2. The parameters in (2) are rarely used in cavity speci-fications. The relationships between these parametersand those in (1a)/(1b) are, however, easily found withthe help of Table I. Examples of cavity parameters
Typical parameters and operating points for some dif-ferent cavities, expressed in the quantities of (2), aregiven in Table II.
IV. PHASOR DIAGRAMS
To better understand the dynamics of the acceleratingmode it is helpful to use phasor diagrams as in Fig. 2.To avoid clutter, the phasor for the cavity field (blue) isshown separately from the phasors that affect its deriva-tive. We will refer to them as the field-decay phasor,the rf-drive phasor (green) and the beam-loading phasor(red).
Remark 5:
Phasor diagrams in the previous literaturetypically show the phasors for the rf drive and the beamloading together with their so-called induced voltages(i.e., their steady-state effects on the cavity field) in a
TABLE II. Parameters for some different cavities in high-energy linacs [11, 12] and an electron storage ring [3, Sec. 17.7].The first group of parameters are inherent to the cavity (al-though γ ext could be tunable), the second group of parametersgives the nominal operating point, and the third group are de-rived quantities of interest. We have approximated γ with 0for superconducting cavities, which is reasonable from an rfsystem perspective. γ / π γ ext / π α A I DC φ b0,lin ∆ E | F g0 | Cavity kHz kHz MV / √ J √ J mA ◦ MeV kWESS RFQ 24 36 3.1 1 . . − a . . . . . −
25 18 2200ESS Medium- β . . . −
15 14 900X-FEL (TESLA) 0 0 .
14 2.9 8 . . ≈ .
016 2.9 5 . . ≈ . . . a The beam loading in a radio-frequency quadrupole is always relative tothe phase of the accelerating mode, i.e., I b ( t ) = I b ( t )e i ( π − φ b, rel ) · e i ∠ A ( t ) .The lumped value φ b, rel is typically not given in RFQ specifications; thevalue − ◦ is an estimate by the author based on the parameters of theindividual RFQ cells [13]. Mode amplitude A ( √ J) Terms of ddt A ( √ J / s)ReIm √ γ ext F g α I b ( − γ + i ∆ ω ) A φ g ReImFIG. 2. Phasor diagrams for visualizing the dynamics of theaccelerating mode in equation (2).
Left:
Phasor for the modeamplitude.
Right:
Phasors that affect the time derivative ofthe mode amplitude; in this figure they sum to zero whichindicates steady-state operation. single diagram [3, 4, 14]. For cavity field control, oneneeds to understand how variations of the phasors F g and I b affect the cavity field. In this regard, the inducedvoltages are of little interest. Leaving the them out anddisplaying the mode amplitude separate from the termsthat affect its time derivative reduces clutter. Remark 6:
In previous literature, the reference phaseis often chosen so that the beam-loading phasor I b isoriented along the negative real axis [4, p. 348]. Thisis reasonable, since after all, the beam phase is the ref-erence relative to which the cavity field should be con-trolled. However, from a field-control perspective (andin particular for linacs), where the objective is to keepthe cavity field close to a setpoint and beam variationsact as disturbances, it is arguably more natural to choosethe reference phase so that the cavity-field phasor lies onthe positive real axis.With this convention we get a nice symmetry in thephasor diagrams for optimally tuned cavities (Fig. 3),with the cavity-field phasor and the rf-drive phasor lying on the real axis. Also, amplitude variations and (small)phase variations of these two phasors correspond to vari-ations of their real and imaginary parts, respectively. Remark 7:
For the particle bunches to experience accel-eration and longitudinal focusing, the beam-loading pha-sor must lie in the second quadrant for circular machinesoperating above transition and in the third quadrant forlinacs and circular machines operating below transition[3, 4, 14].
V. POWER-OPTIMAL COUPLING ANDDETUNING
In this section we compute the detuning ∆ ω and thecoupling γ ext that minimize the rf drive power duringsteady-state operation. These are standard calculations[1–4], but we go through them to show what they lookwith the parameterization (2) and because we need theresults in the next section. Note that these calculationsthat use (2) are arguably more clear than those in theprevious literature.Assume that the nominal mode amplitude is given by A = A > I b0 . The corresponding stationary rf drive F g0 satisfies0 = ( − γ + i ∆ ω ) A + p γ ext F g0 + α I b0 , (5)which gives the rf drive power P g0 = | F g0 | = 12 γ ext (cid:12)(cid:12)(cid:12) ( − γ + i ∆ ω ) A + α I b0 (cid:12)(cid:12)(cid:12) . We see that the power consumption P g0 is minimized bymaking the imaginary part of the expression within theabsolute value zero by choosing the detuning as∆ ω ⋆ = − α · Im I b0 A . (6)If the cavity is optimally tuned, then (at steady-state)the rf-drive phasor lies on the positive real axis, andthe imaginary parts of the decay phasor and the beam-loading phasor have equal magnitudes but opposite signs.It can be seen that Figs. 3 and 4 correspond to optimallytuned cavities, but that Fig. 2 does not.With power-optimal detuning we have P g0 (cid:12)(cid:12)(cid:12) ∆ ω =∆ ω ⋆ = 12 γ ext (cid:16) ( γ + γ ext ) A − α I b0 (cid:17) . Recall Remark 6.
Mode amplitude A ( √ J) Terms of ddt A ( √ J / s) √ γ ext F g α I b ( − γ + i ∆ ω ) A FIG. 3. Phasor diagram for a superconducting cavity that isoptimally tuned and optimally coupled. − γ A − γ ext A i ∆ ω A √ γ ext F g α I b Terms of ddt A ( √ J / s)FIG. 4. Phasor diagram with the terms of the time deriva-tive of the cavity field. The considered cavity is optimallytuned (∆ ωA = − ( α/ I b0 ), optimally coupled ( γ ext A = γ A − ( α/ I b0 ), and normal conducting ( γ > Minimizing this expression with respect to γ ext , gives thepower-optimal coupling coefficient γ ⋆ ext = γ − α I b0 A . (7)Thus, given A and I b0 , the minimal power consump-tion equals P ⋆ g0 = 2 γ A − αA · Re I b0 . That is, all energy in the forward wave is either dissipatedin the cavity walls or transferred to the particle beam—no power is wasted in the reverse wave. The total decayrate, assuming optimal coupling, is given by γ = 2 γ − α I b0 A . (8) Remark 8:
We have found that it sometimes gives intu-ition to think of the second term in (7) as the decay rate We wish to minimize 2 f ( x ) = ( ux + v ) /x = u x + 2 uv + v /x wrt x >
0. Differentiating gives 2 f ′ ( x ) = u − v /x , from whichwe find the optimal point x ⋆ = v/u , at which f ( x ⋆ ) = 2 vu . of the cavity field due to beam loading (at the nominaloperating point). This motivates the definition γ beam = γ beam ( A , I b0 ) := − α I b0 A . (9)Using (9), we can write (7) more intuitively as γ ⋆ ext = γ + γ beam . Remark 9:
There are two different conventions for thesynchronous phase φ b0 , i.e., the nominal phase betweenthe particle bunches and the accelerating mode [2]. Forlinacs and circular electron machines it is conventionallydefined so that φ b0,lin = π − ∠ I b0 [1, 3, 4], which givesthe well-known expressions∆ ω ⋆ = − αI b sin φ b0,lin A , γ ⋆ ext = γ + αI b cos φ b0,lin A . For circular proton machines [2, 14], the convention issuch that φ b0,circ = 3 π/ − ∠ I b0 , which gives∆ ω ⋆ = αI b cos φ b0,circ A , γ ⋆ ext = γ + αI b sin φ b0,circ A . Remark 10:
For pulsed linacs, the value of γ ext that min-imizes the overall power consumption is somewhat largerthan γ ⋆ ext since this gives a shorter filling time. This isparticularly important when the pulses are short com-pared to the filling time. It is also better to choose γ ext larger than γ ⋆ ext if there are significant detuning varia-tions during the flat-top (e.g., from microphonics) [9]. Remark 11:
The coupling factor β = Q /Q ext = γ ext /γ is commonly used in previous derivations of optimal cou-pling [3, 4, 9, 10]. For perfectly superconducting cavities( β = ∞ ), many expressions in those derivations are ill-defined. The derivation in this section avoids that unaes-thetic feature. Remark 12:
The detuning angle ψ = tan − (∆ ω/γ ) isoften used for describing the steady-state response of theaccelerating mode. Note that for the transfer function P a ( s ) in (12) we have P a (0) = cos ψ · e iψ . VI. NORMALIZED CAVITY DYNAMICSA. Normalization
Requirements on cavity-field errors, and specificationson amplifier ripple and beam-current ripple are typicallygiven in relative terms, i.e., on the form x % and y ◦ . Bynormalizing the dynamics from disturbances to field er-rors (of the accelerating mode), it is easy to compute therelative field errors that result from relative disturbances.For control design, it is convenient if the static gainfrom control action to mode amplitude is one.For these reasons, we introduce the following normal-ized phasors for the cavity field, rf drive, and beam load-ing, colored according to Fig. 2, a := 1 A A (10a) f g := 1 γA p γ ext F g (10b) i b := 1 γA α I b . (10c)Scaling equation (2) by 1 /A gives˙ a = ( − γ + i ∆ ω ) a + γ ( f g + i b ) . (11)The transfer function from f g and i b to a is given by P a ( s ) := γs + γ − i ∆ ω , (12)where the subscript a indicates the accelerating mode. Remark 13:
The relative beam-loading parameter Y in[15] corresponds to | i b0 | . B. Relations at nominal operating point
Consider steady-state operation at some nominal op-erating point ( a = 1 , f g0 , i b0 ). For an optimally tuned cavity it follows from (6) that γ Im i b0 + ∆ ω = 0 . (13)For an optimally coupled cavity, it follows from (7) that − ≤ Re i b0 ≤ . (14)For an optimally tuned and optimally coupled cavity wehave that f g0 = 1 − i ∆ ω/γ − i b0 is real, and that1 ≤ f g0 ≤ . (15)For a superconducting cavity ( γ = 0) that is optimallytuned and coupled, we have that Re i b = − f g = 2. C. Dynamics around nominal operating point
Cavity field stability is typically evaluated aroundsome nominal operating point. For this reason it is mean-ingful to introduce the normalized field error zzz through zzz = a − . It is clear that a small field error zzz = z re + iz im approx-imately corresponds to an amplitude error of z re ·
100 %and a phase error of z im rad.In the case of an ideal amplifier, we would have f g = f g0 + ˜ f g where ˜ f g corresponds to control action from the TABLE III. Normalized parameters for the cavities in Ta-ble II. γ/ π f g0 | i b0 | φ b Cavity kHz − − ◦ ESS RFQ 60 1.1 0.2 − − β − ≈ ≈ field controller. However, due to variations of the ampli-fier’s gain and phase shift, denoted by ˜ g amp and ˜ θ amp , wehave f g = (1 + ˜ g amp )e − i ˜ θ amp ( f g0 + ˜ f g ) . (16)Assuming that the variations ˜ g amp and ˜ θ amp are small,and introducing d g := ˜ g amp − i ˜ θ amp , it follows from (16)that f g ≈ (1 + d g )( f g0 + ˜ f g ) ≈ f g0 + ˜ f g + f g0 d g . (17)Similarly, relative beam loading variations d b affect i b according to i b = (1 + d b ) i b0 . Plugging these expressions into (11) and ignoringsecond-order terms give˙ zzz = ( − γ + i ∆ ω ) zzz + γ (cid:16) ˜ f g + f g0 d g + i b0 d b (cid:17) . (18)We see that the transfer function from relative distur-bances d g and d b to relative field errors zzz are given by P d g → z ( s ) = f g0 P a ( s ) , (19a) P d b → z ( s ) = i b0 P a ( s ) , (19b)where P a ( s ) is defined in (12). In these equations, thenominal phasors f g0 and i b0 act as complex-valued coef-ficients that quantify the impact of relative disturbances. Example:
The normalized cavity parameters γ , f g0 , and i b0 for the cavities in Table II are shown in Table III. Remark 14:
In superconducting cavities, the detuning∆ ω may vary due to microphonics and Lorenz-force de-tuning. If we denote the detuning variations by g ∆ ω wecan define the normalized detuning variations d ∆ ω := g ∆ ω/γ . If the detuning variations are small, they canbe included as a term iγ · d ∆ ω on the right-hand sideof (18). The transfer function from normalized detuningvariations to field errors is given by P d ∆ ω → z ( s ) = iP a ( s ). Remark 15:
The disturbances d g and d b tend to havea certain directionality. For example, rf amplifiers suchas klystrons, typically have more phase variations than ( − γ + i ∆ ω ) a γ i b γ f g γ i b0 d b γ f g0 d g FIG. 5. Visualization of phase variations of the rf drive ( d g purely imaginary) and amplitude variations of the beam cur-rent ( d b purely real). The effect on the accelerating cavitymode is given by filtering the illustrated variations throughthe transfer function P a ( s ) in (12). amplitude variations, corresponding to that d g is domi-nantly imaginary. Beam-current ripple on the other handaffects the magnitude of i b (corresponding to a real d b ).The directionality of these disturbances are readily vi-sualized as in Fig. 5, using the phasor diagrams of Sec-tion IV. Synchrotron oscillations in circular acceleratorscorrespond to an imaginary d b . VII. SUMMARY
We have proposed an energy-based parameterizationof a cavity’s accelerating-mode dynamics. The proposedparameterization avoids many problems of equivalent-circuit based parameterizations and is helpful for under-standing the impact of cavity parameters on the rf systemand field control loop. We have also provided a normal-ized model that is suitable for field control design.
ACKNOWLEDGMENTS
The author thanks Daniel Sjberg, Larry Doolittle, andBo Bernhardsson for helpful comments and suggestions.The author is a member of the ELLIIT Strategic Re-search Area at Lund University.
APPENDIX: DERIVATION OF EQUATION (2)
We show how equation (2) follow from conservation ofenergy together with linearity and time-reversibility ofMaxwell’s equations. The considered system, togetherwith the notation that will be used, are shown in Fig. 1.Note that the derivation in this appendix is not in-tended to prove a new result; equation (2) simply followsfrom (1a)/(1b) by using Table I. The derivation is in-stead included for an easy-to-follow connection betweenthe model (2) and the physical cavity–waveguide system.We start by going through Haus’ derivation for the dy-namics of a waveguide-coupled cavity [7, Sec. 7.2], usingthe notation of this paper. Then we show how the impactof beam loading can be included. It is assumed that: (1) changes of the mode shape due to resistive losses, the ex-ternal coupling, and detuning variation can be neglected;(2) the slowly-varying envelope approximation holds. Formodeling of multiport cavities, see [16].
1. Maxwell’s equation for the electromagnetic field
We start by considering a lossless cavity without beamconnected to a waveguide, and later introduce losses andbeam loading in sections 4 and 6. From Maxwell’s equa-tions, we have that the following equation holds for theelectric field E = E ( r , t ) in the cavity and the waveguide ∇ E − ǫ µ ∂ ∂t E = 0 , (20)where ε and µ are the permittivity and permeability offree space.
2. Mode expansion of the cavity field
Assume for a moment that the cavity is not coupled tothe waveguide. The electric field in the cavity can thenbe expanded as a sum of orthogonal eigenmodes E k E ( r , t ) = ∞ X k =0 e k ( t ) E k ( r ) (21)where the mode amplitudes e k ( t ) evolve independentlyaccording to d dt e k ( t ) = − ω k e k ( t ) . (22)
3. Baseband dynamics of the accelerating modein a lossless cavity
We will only consider the specific mode used for par-ticle acceleration. When necessary, we will label relatedquantities with a subscript a . To simplify the exposition,and keep with the spirit of the paper, we will work withthe complex envelope A of the accelerating mode (rel-ative to some phase reference with frequency ω rf ); i.e., e a ( t ) = Re { A ( t )e iω rf t } . We will also assume that themode amplitude is normalized so that | A | equals the en-ergy stored in the mode ( A has units √ J). From (22) itfollows that ddt A = i ∆ ω A (23)where ∆ ω := ω a − ω rf .
4. Waveguide coupling
Now, assume that the cavity is connected to a waveg-uide by a coupling port as in Fig. 1. An incident forwardwave in the waveguide will excite the accelerating modethrough the coupling port, but energy will also escapethe cavity through the port and propagate away in a re-verse wave (Fig. 1). Denote the complex-envelopes (withrespect to ω rf ) of the forward and reverse waves by F g and R g , and assume them normalized so that | F g | isthe power of the forward wave ( F g has units √ W) andsimilarly for R g .Due to the linearity of Maxwell’s equations, we have ddt A = i ∆ ω A − γ ext A + κ g F g (24)where γ ext is the rate at which the cavity field decaysthrough the coupling port, and κ g is a, possibly complex-valued, parameter that quantifies the effect of the forwardwave on the cavity field. Not surprisingly, γ ext and κ g are related and we next derive how.
5. Relation between γ ext and κ g Throughout this subsection we consider the particularsolution to (24) for t ≥ A (0) = 1, with ∆ ω = 0 and F g ( t ) ≡
0. It isclear that the solution is given by A ( t ) = e − γ ext t , t ≥ . (25)Recall that the energy stored in the accelerating modeis | A | and hence changes by ddt | A | = − γ ext e − γ ext t .Due to conservation of energy, this power is carried awayby the reverse wave, hence | R g ( t ) | = 2 γ ext e − γ ext t . (26)If E ( r , t ) is a solution to (20), valid in the cavityand the waveguide, then so is the time-reversed solution E r ( r , t ) = E ( r , − t ).Time-reversal of the particular solution considered inthis subsection gives a solution where the evolution of theaccelerating mode is given by A r ( t ) = A ( − t ) = e γ ext t , t ≤ , (27)the reverse wave satisfies R g r ( t ) = F g ( t ) ≡
0, and theforward wave satisfies | F g r ( t ) | = | R g ( − t ) | . From (26) wethen have that F g r ( t ) = e iφ · p γ ext e γ ext t (28)for some phase φ . The variables of the original solutionand time-reversed solution are illustrated in Fig. 6.Recalling that ∆ ω = 0, and plugging (27) and (28)into (24) gives γ ext e γ ext t = − γ ext e γ ext t + κ g e iφ · p γ ext e γ ext t , from which it follows that κ g = e − iφ · p γ ext . The reference phase for the forward wave F g can bechosen freely; choosing it such that φ = 0 gives κ g = p γ ext . Original solutionundef. 01 t A Time-reversed solutionundef.01 t A r t | F g | t | F g r | t | R g | t | R g r | FIG. 6. The particular solution considered in Sec. 5.
6. Beam loading
Consider a beam, i.e., a train of charged bunches,traversing the cavity. Assume that the bunches are regu-larly spaced in time by an integral number of rf periods.Let the beam be modeled by the complex signal I b whosemagnitude | I b | equals the dc current of the beam. Thephase of I b is defined so that ∠ I b = − π corresponds tomaximum acceleration (energy gain) from the nominalfield of the accelerating mode (corresponding to ∠ A = 0).Note that I b is allowed to vary slowly.Define the cavity–beam-coupling parameter α of theaccelerating mode so that the following equality holds(in [4, Ch. 2] it is shown that such an α exists)power to beam from acceler-ating mode = − Re { α I ∗ b A } . (29)The cavity–beam-coupling parameter of the accelerat-ing mode is real and non-negative due to the definition ofthe I b . For a general mode k , the cavity–beam-couplingparameter is in general complex and the α on the right-hand side of (29) should be replaced by α ∗ k .The bunch train induces an electromagnetic field inthe cavity, corresponding to a term − µ d J /dt , where J is current density, on the right-hand side of (20). Thiseffect is linear and corresponds to a term c b I b , where c b is a complex coefficient, on the right-hand side of (23).Assuming for a moment that ∆ ω = 0, we have d A dt = c b I b . Taking the time derivative of the energy in the accelerat-ing mode and using this expression we get that ddt | A | = 2Re { c ∗ b I ∗ b A } . (30)Conservation of energy gives that (29) and (30) sum tozero. Since this holds for all I b it follows that c b = α/
7. Putting the pieces together
By combing the results from the two preceding sectionsand adding a term − γ A for resistive losses (assumingthat this does not significantly change the mode shape)we arrive at d A dt = ( − γ + i ∆ ω ) A + p γ ext F g + α I b (31)where γ = γ + γ ext . This is exactly (2).
8. The reverse wave
As in [7, (7.36)] we may derive an expression for theenvelope R g of the reverse wave. From the linearity ofMaxwell’s equations we know that the reverse wave de-pends linearly on the forward wave and the cavity field R g = c F F g + c A A , where c F and c A are complex constants. We alreadyknow from (25) and (26) that | c A | = √ γ ext . Since weare free to choose the reference phase for R g we will take c A = √ γ ext .Next, we derive an expression for c F . Conservation ofenergy gives that | F g | − | R g | = ddt | A | , (32) and from (31) (with I b = 0) it follows that ddt | A | = − γ ext | A | + p γ ext (cid:16) A ∗ F g + F ∗ g A (cid:17) . (33)Putting (32) equal to (33), and then substituting A =( R g − c F F g ) / √ γ ext gives | F g | − | R g | = − γ ext | A | + p γ ext (cid:16) A ∗ F g + F ∗ g A (cid:17) = − (cid:16) | R g | − c F R ∗ g F g − c ∗ F F ∗ g R g + | c F | | F g | (cid:17) + (cid:16) R ∗ g F g − c ∗ F | F g | + F ∗ g R g − c F | F g | (cid:17) . From this equation it follows that | (1 + c F ) F g − R g | = | R g | . For this equality to hold for all F g and R g , we must havethat c F = − R g = − F g + p γ ext A . [1] Thomas Schilcher, Vector Sum Control of Pulsed Accel-erating Fields in Lorentz Force Detuned SuperconductingCavities , Ph.D. thesis, University of Hamburg, Hamburg,Germany (1998).[2] Joachim T¨uckmantel,
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