STABILITY CONSIDERATIONS

Transcription

1 Abstract The simple theory describing the stability of an RF system with beam will be recalled together with its application to the LEP case. The so-called nd Robinson stability limit can be pushed by a large factor using RF feedback on the vector sum of individual cavity signals, and should not be a limitation for LEP. This technique also reduces the effect on the beam of cavity oscillations. Finally the significance of the measured Q s spectrum will be discussed. ROBINSON LIMITS In a classical accelerator theory paper (K.W. Robinson, 964) [] the limits of stability of an accelerating cavity with beam loading were given. The domain of validity of the analysis is limited to cases where the entire RF system can be modelled by a single RLC circuit, and to beam-cavity interactions involving only frequencies much below the revolution frequency. In particular the broad-band impedance contribution from the RF system is not taken into account. We start with the usual cavity model (Fig. ), in which the cavity itself, as seen from the accelerating gap, is represented by a lumped-elements RLC circuit. The beam is a pure current source I B (assuming that the beam energy is much larger than the cavity voltage), and the power generator a second current source I. In this model the generator impedance, transformed to the accelerating gap, has been included in the shunt resistance R. The vector diagram of Fig, at the RF frequency represents the steady state situation of a single cavity (or if all cavities are in phase and identical). The cavity voltage V results from the total current IT I+ IB flowing into the parallel RLC circuit impedance, characterized by the on-tune current I0 V/ R and the detuning angle φz ( tan φz Q f / frf, where Q is the cavity quality factor, including generator, f the cavity detuning with respect to the RF frequency f RF ). The phase of the beam current I B with respect to V is simply φb π B is the synchronous phase angle determined by independent machine parameters (rate of rise of magnetic field and (or) synchrotron radiation). Our free parameter is the generator phase angle φ L (phase between I and V ) which can be adjusted via the cavity tuning. The normal operating condition is φ L 0, which corresponds to a resistive load to the power generator. STABILITY CONSIERATIONS aniel Boussard, CERN, eneva, Switzerland Fig. Lumped circuit equivalent of accelerating cavity Fig. Steady state vector diagram THE PEERSEN MOEL In the following we shall follow the analysis of F. Pedersen []. The variables are the phase and amplitude modulations of the three vectors I, V and IB. They are related by the cavity transfer functions and B from generator and beam respectively as shown in Fig. 3. For simplicity we now only consider the case of short bunches, so that the amplitude of I B is constant, (I B I C ) and the corresponding variable disappears. Phase and amplitude modulations can be represented by their two rotating vectors at the modulation frequency. The cavity transfer functions from modulation of I T to modulation of V are therefore given by: 08

2 Fig. 3 The Pedersen model pp aa pa ap Zs+ jωrf Zj ( ωrf ) Zs+ jωrf Zj ( ωrf ) Zs jωrf + Zj ( ωrf ) () Zs ( jωrf ) Zj ( ωrf ) where pp, aa are the transfer functions from phase modulation to phase modulation, or amplitude modulation to amplitude modulation, and pa, ap are the transfer functions from phase modulation to amplitude modulation and vice versa. The cavity impedance Z(s) is given by: σrs Zs () () s + σs+ ωr where σ ω r / Q is the half cavity bandwidth (in rad/s) and ω r the cavity resonant frequency. ω r is related to the detuning angle by: ω r ω RF + σ tan φ z (3) Combining equations () and () one finds: σ ( + tan φz )+ σs pp (4) s + σs+ σ + tan φz σtan φ s z pa s + σs+ σ + tan φz The final transfer functions from IB and I to V are obtained by projection of the modulations of IB or I on IT according to: (5) I I I I pp pp cos T, + pa sin IT, I (6) IT IT B B and similar relations for pa, pp and pa, etc. Finally one obtains the complete transfer functions: Y Y s σ ( + tan φ z + ( sin φ B tan φ z cos φ B ))+ σ( + sin φ B ) pp Y B z B z Y B s pa σ ( cos φ + tan φ sin φ )+ σ ( tan φ cos φ ) Y( ( s B σ tan φ z cos φ B sin φ B ) σsin φ ) B pp (7) Y( B σ ( tan φ z sin φ B + cos φb)+ σcos φbs) pa s + σs+ σ ( + tan φz) Y IB/ Iο aa pp ap pa. The beam transfer function (in other words the effect of phase modulation of the RF voltage on the beam phase) is that of an undamped resonator B ω s ( s + ωs ), as we neglect here the natural synchrotron radiation damping. The effect of amplitude modulation on the dipole mode (beam phase modulation) is the same, except for the additional factor tan φ B, which disappears in the case of no acceleration (φ B 0 or π). This is represented in Fig.3. Let us now look at the stability of the RF system without electronic loops, i.e. in the case where the RF generator is unmodulated, either in phase, or in amplitude. Figure 3 simplifies considerably and we are left with only its right side, leading to the characteristic equation: B ( B pp + tan φb B pa) 0 (8) which can be written in polynomial form: s 4 + σ s 3 + ( ω s + σ ( + tan φ z) ) s + σω s s + + σωs( + tan φz Ytan φz / cos φb) 0 (9) The Routh stability conditions [3]: aa a 4 > 0 a3 and: aa a3 a a4 a3 a0 > 0, where a 0, a, a, a 3, a 4 are the coefficients of the polynomial equation in increasing order, applied to (9) lead to the two inequalities: Y sin φz < (0) cos φb 09

3 and tan φ z > 0 () which are referred to as the Robinson stability limits []. The second one (equation ) is independent of intensity (low intensity limit), whereas the first puts a threshold on the beam current (proportional to Y). The low intensity limit has often been described in the literature on bunched beam instabilities. The RF cavity is a narrow band resonator whose resistive part may be different for the two synchrotron side bands at f RF + f s and f RF - f s. Below transition, stability of the n 0 mode (all bunches in phase) is achieved if R(f RF + f s ) > R(f RF - f s ), which also corresponds to a positive detuning, or φ > 0 (Fig. 4). To interpret physically equation (0) (high intensity limit), let us consider the beam induced voltage VB ZIB and the generator induced voltage V ZI, whose sum VB+ V is simply V. The corresponding vector diagram (Fig. 5) is the same as the current diagram of Fig., IB+ I IT, but shifted in phase by φ z. If the quantity tan( φ z φl) is evaluated at the stability limit: Y cos φb / sin φ z () using the relation: tan φ cos tan φ z Y φb L (3) + Ysin φb Fig. 4 Low intensity Robinson stability (R + > R - for γ < γ T ) Fig. 5 Vector diagram showing V VB+ V obtained from Fig., one easily finds: tan ( φz φl) tan φb (4) showing that the vectors V and IB are in opposition. The stability condition (0) can therefore be interpreted as the limit where the beam sits on the crest of the generator induced voltage. For the coherent dipole motion of the bunch, only this voltage can provide a restoring force (at the instability frequency, s 0, the beam induced voltage simply follows the bunch motion and cannot contribute). Therefore when the vector combination is such that the beam reaches the crest of the generator driven voltage, longitudinal focusing (or stability) is lost for the dipole mode. The stability limits (0) and () are shown on the diagram of Fig. 6, where the dashed areas are forbidden. The curve tan φ L 0 or Y tan φ z /cos φ B, corresponding to a resistive load to the power source is also shown on this diagram. Combining (), (3) and the following equation obtained from Fig. : I I0( + Ysin φ B) / cos L (5) one finds, at the threshold current: I I0 0

4 as plotted in Fig. 8. Comparing with Fig. 6, the effect of the phase loop is to remove the low intensity Robinson limit (φ z > 0). In other words, the strong damping imposed by the phase loop at low beam currents, overrides the small antidamping which would prevail at φ z < 0. Fig. 6 Robinson stability limits for the optimum power transfer situation (φ L 0). In other words, the power delivered to the beam is just equal to V R at the current limit. In the case of a currentsource-type generator (tetrode, for instance) this corresponds to the power lost in the cavity walls and the tube. If a matched generator is used (klystron and circulator for instance) in conjunction with a lossless cavity (superconducting), when all power is transferred to the beam (matched generator I I 0 ), one is sitting just at the stability limit [4]. STABILITY LIMITS WITH PHASE LOOP In proton machines a phase loop is very efficient for stabilizing the RF system at moderate intensities. To evaluate its effect, the diagram of Fig. 3 must be completed by the phase loop path: measurement of the phase difference between I B and V and correction via the phase of the generator current I (Fig. 7). For simplicity assume φ B 0, short bunches, and an idealized transfer function: Cp ω p / sfor the Voltage Controlled Oscillator. The characteristic equation can now be written: s + ωs s + σs+ σ + tan φz ( ) ω σ φ ω ( σ s Y z ps ( + φ z Y φ z) + σ s) tan tan tan. (6) Applying again Routh's criterion to this 4th-order equation gives the following sufficient stability conditions for a large loop gain: φz > 0 Y < (7) sin φz φz < 0 Y < tan φz (8) On the other hand, the high current limit is not affected by the phase loop, as shown by equation (7). This can also be understood in the following way: the high current limit corresponds to s 0, for which the overall phase loop transfer function ω / s s / ω + s ( p ) ( s ) vanishes. The phase loop becomes inefficient when the frequency of the dipole mode approaches zero, i.e. when the high current Robinson limit is reached. Fig. 7 Phase loop diagram (φ B 0) Fig. 8 Robinson stability limits with phase loop (φ B 0)

5 3 EVOLUTION OF THE ROOTS OF THE CHARACTERISTIC EQUATION Let us return to the characteristic equation (9), valid in the case of no loops. For a vanishing beam current (Y 0), the equation has two purely imaginary roots s± jω s and two complex roots s σ( ± jtan φz) with a negative real part. Obviously the imaginary roots correspond to the undamped synchrotron oscillation and the two others to the cavity damping (including detuning). It is interesting to evaluate how these roots move, especially the two imaginary ones when the beam current is increased. Assuming Y small and looking for solutions of the form s jω s ( + ε) with ε <<, we obtain: σ Ytan φ ε z (9) cosφ B ωs jσωs σ ( + tan φz) If the cavity were to stay on tune (φ z 0) at the RF frequency, there would be no shift of ω s, either real or imaginary. In other words the coherent dipole frequency would be independent of beam current. This can easily be explained by the classical instability theory, which states that the shift of coherent frequency ω s is proportional to the sum + p IbZ p, where Z is the impedance and p the mode number (positive and negative). For a cavity on tune, centred at f RF, this sum vanishes irrespective of I b. To evaluate equation (9), assume for simplicity φ B 0 (this is the case at LEP injection energy), σ << ω s (this is the case for the LEP SC cavities) and optimum cavity detuning (φ L 0). We obtain: 3 ω σ σ ε s Y + j Y (0) ω s ωs ωs The first term corresponds to an increase of the coherent frequency, proportional to the square of the beam current. Its magnitude is small; for LEP typical parameters at injection σ/ω s 0. and Y, we obtain ω s /ω s 0.5%. The second term provides damping; with the above LEP parameters, the damping time is typically 50 ms. When increasing the beam current, one of the other two roots of the characteristic equation [s - σ( ± jtanφ z ) for Y 0] becomes purely real and moves along the negative real axis towards the origin. It is that particular root which gives the high-intensity Robinson limit, when it reaches the origin; it is not that the coherent frequency decreases and becomes zero at the threshold. In fact the two complex conjugate roots [s ± jω s ( + ε)] still exist at the threshold. 4 VECTOR SUM RF FEEBACK The classical solution to avoid the high-intensity Robinson limit is to use RF feedback around the amplifier as sketched in Fig. 9 Fig. 9 The RF feedback principle An additional current source I F, proportional to the cavity voltage V is put across the RLC cavity equivalent circuit. Its effect is that of an additional shunt resistance ( I F proportional to V ) whose value is inversely proportional to the loop gain H. Seen from outside the cavity-amplifier system looks like a cavity with a lower shunt resistance [R R/(+H)] a large bandwidth [σ σ(+h)], a smaller detuning angle [ tan φ z tan φz ( + H) ] and a smaller beam loading factor Y Y. + H The new high-intensity Robinson limit becomes: sin φ Y z < () φb sin φ Y z + tanφ z < cos B () φ ( + H) + tanφz The threshold is increased by the factor ( + ) + ( H tan φz) ( + tan φ z) which scales approximately like H for H >> and tanφ z unity. In LEP, RF feedback is being implemented on each klystron, with a loop gain of about 6 db. When all units are equipped, the high-intensity Robinson limit will be pushed away by more than an order of magnitude, much beyond many other intensity limits. Therefore, it cannot be considered as a limitation for LEP. The coherent mode frequency shift remains unchanged, but the damping is increased by the factor ( + H) (equation 0). In LEP, where one klystron drives eight cavities, the cavity signal is made by reconstructing the total voltage seen by the beam when traversing the eight cavities, with a vector sum network (Fig. 0). The accuracy of the

6 vector sum reconstruction limits the useful gain of the feedback. lobal voltage control RF ref. in Mod. Power/current feedback K C C C 3 C 4 C 5 Σ configuration of the RF system (for instance, there is more noise off resonance with vector sum feedback on) and therefore the information given by the Q s display must be taken with precaution. A possible noise source might also be the fluctuations of the bending magnetic field. espite the filtering effects of the eddy currents in the vacuum chamber this noise source may not be negligible. Phase loop C 6 C 7 C 8 Σ φ Fig. 0 Simplified block diagram of the vector-sum feedback Σ Fig. A typical Q s spectrum display The vector sum RF feedback can also be regarded as a voltage follower which keeps the actual voltage seen by the beam during its traversal of the eight cavities, almost equal to the reference voltage, despite outside perturbations. These are typically due to mechanical cavity oscillations and to tuning offsets applied individually to each cavity in order to avoid ponderomotive instabilities. 5 SOME REMARKS ON THE Q S SPECTRUM OBSERVE IN LEP A typical Q s spectrum is shown in Fig., from which we infer a synchrotron frequency of about.5 khz. The observed noise spectrum is the result of a noise excitation on the beam in frequency bands around nfrev ± fs where f rev is the revolution frequency. Looking at the coherence between Q s signals of bunches separated by a quarter-turn (the signals are almost completely correlated) one concludes that the noise source is almost entirely in a frequency band around f s (dipole mode, n 0). The magnitude of the noise excitation can be measured far off the beam resonance, e.g rad Hz or -9.7 dbc/ Hz at 900 Hz (Fig. ). This corresponds very well to the typical measured noise on a cavity voltage (-90 dbc/ Hz ) (Fig. ) and shows that the noise present on the RF system is probably the dominant source of excitation of the Q s spectrum. Unfortunately this source is not white (Fig. ); it also depends on the Fig. Noise spectrum of a superconducting cavity in LEP REFERENCES [] K.W. Robinson, CEA Report No. CEAL0 (Feb. 964). [] F. Pedersen, Beam Loading Effects in the CERN PS Booster, PAC Conference, Washington 975. [3] E.J. Routh, A Treatise on the ynamics of a System of Rigid Bodies, published by McMillan, London, 977. [4]. Boussard, esign of a Ring RF System, CAS RF Engineering for Particle Accelerators, Oxford, 99, CERN 9-03 (99). 3