Longitudinal stability for the Super-PEP Super-B Factory Workshop in Hawaii April 20-22, 2005 Dmitry Teytelman Super-B Factory Workshop in Hawaii April 21, 2005
Outline I. The scope of this talk II. Fundamental impedance of RF cavities and longitudinal instabilities III. Parameter selection procedure IV. Evaluation of cavity options V. Summary Super-B Factory Workshop in Hawaii April 21, 2005
The scope of this talk Will discuss only the fundamental impedance of the RF cavities and its effect on the longitudinal coupled-bunch stability. Why? • In the existing B Factories the fundamental impedance drives the fastest growing modes • While one can work on damping HOM impedances the fundamental impedance cannot be reduced except at the initial machine design stage. • Very high beam currents in the Super-PEP design mean high beam loading with attendant high detuning ➔ likely high fundamental-driven growth rates Not discussed: • Longitudinal • Dipole coupled-bunch instabilities driven by the HOMs • Higher-order intra- and inter-bunch instabilities • Transverse • Dipole coupled-bunch instabilities due to the resistive wall and the HOMs Super-B Factory Workshop in Hawaii April 21, 2005
Fundamental impedance and coupled-bunch instabilities The growth rate of 2000 eigenmode -1 is proportional to the 1500 ℜ (Z), k Ω difference between the real 1000 parts of the impedance at 500 – + and ω rf ω rev ω s + + ω rf ω rev ω s 0 −1.5 −1 −0.5 0 0.5 1 1.5 Frequency offset from ω rf , revolution harmonics When the cavity is at Effective driving impedance, k Ω 2000 resonance that difference is very small 1000 However with increasing 0 beam current the cavity −1000 center frequency is −2000 detuned below the RF −1.5 −1 −0.5 0 0.5 1 1.5 Eigenmode number frequency causing larger and larger asymmetries When the detuning is comparable to the revolution frequency one needs RF feedback to reduce the effective impedance presented to the beam Super-B Factory Workshop in Hawaii April 21, 2005
Fundamental impedance and coupled-bunch instabilities The growth rate of 2000 eigenmode -1 is proportional to the 1500 ℜ (Z), k Ω difference between the real 1000 parts of the impedance at 500 – + and ω rf ω rev ω s + + ω rf ω rev ω s 0 −1.5 −1 −0.5 0 0.5 1 1.5 Frequency offset from ω rf , revolution harmonics When the cavity is at Effective driving impedance, k Ω 2000 resonance that difference is very small 1000 However with increasing 0 beam current the cavity −1000 center frequency is −2000 detuned below the RF −1.5 −1 −0.5 0 0.5 1 1.5 Eigenmode number frequency causing larger and larger asymmetries When the detuning is comparable to the revolution frequency one needs RF feedback to reduce the effective impedance presented to the beam Super-B Factory Workshop in Hawaii April 21, 2005
Fundamental impedance and coupled-bunch instabilities The growth rate of 2000 2000 eigenmode -1 is proportional to the 1500 1500 ℜ (Z), k Ω ℜ (Z), k Ω difference between the real 1000 1000 parts of the impedance at 500 500 – + and ω rf ω rev ω s + + ω rf ω rev ω s 0 0 −1.5 −1.5 −1 −1 −0.5 −0.5 0 0 0.5 0.5 1 1 1.5 1.5 Frequency offset from ω rf , revolution harmonics Frequency offset from ω rf , revolution harmonics When the cavity is at Effective driving impedance, k Ω Effective driving impedance, k Ω 2000 2000 resonance that difference is very small 1000 1000 However with increasing 0 0 beam current the cavity −1000 −1000 center frequency is −2000 −2000 detuned below the RF −1.5 −1.5 −1 −1 −0.5 −0.5 0 0 0.5 0.5 1 1 1.5 1.5 Eigenmode number Eigenmode number frequency causing larger and larger asymmetries When the detuning is comparable to the revolution frequency one needs RF feedback to reduce the effective impedance presented to the beam Super-B Factory Workshop in Hawaii April 21, 2005
Fundamental impedance and coupled-bunch instabilities The growth rate of 2000 eigenmode -1 is proportional to the 1500 ℜ (Z), k Ω difference between the real 1000 parts of the impedance at 500 – + and ω rf ω rev ω s + + ω rf ω rev ω s 0 −1.5 −1 −0.5 0 0.5 1 1.5 Frequency offset from ω rf , revolution harmonics When the cavity is at Effective driving impedance, k Ω 2000 resonance that difference is very small 1000 However with increasing 0 beam current the cavity −1000 center frequency is −2000 detuned below the RF −1.5 −1 −0.5 0 0.5 1 1.5 Eigenmode number frequency causing larger and larger asymmetries When the detuning is comparable to the revolution frequency one needs RF feedback to reduce the effective impedance presented to the beam Super-B Factory Workshop in Hawaii April 21, 2005
Fundamental impedance and coupled-bunch instabilities The growth rate of 2000 2000 eigenmode -1 is proportional to the 1500 1500 ℜ (Z), k Ω ℜ (Z), k Ω difference between the real 1000 1000 parts of the impedance at 500 500 – + and ω rf ω rev ω s + + ω rf ω rev ω s 0 0 −1.5 −1.5 −1 −1 −0.5 −0.5 0 0 0.5 0.5 1 1 1.5 1.5 Frequency offset from ω rf , revolution harmonics Frequency offset from ω rf , revolution harmonics When the cavity is at Effective driving impedance, k Ω Effective driving impedance, k Ω 2000 2000 resonance that difference is very small 1000 1000 However with increasing 0 0 beam current the cavity −1000 −1000 center frequency is −2000 −2000 detuned below the RF −1.5 −1.5 −1 −1 −0.5 −0.5 0 0 0.5 0.5 1 1 1.5 1.5 Eigenmode number Eigenmode number frequency causing larger and larger asymmetries When the detuning is comparable to the revolution frequency one needs RF feedback to reduce the effective impedance presented to the beam Super-B Factory Workshop in Hawaii April 21, 2005
PEP-II low-level RF feedback: impedances and growth rates Two feedback loops are used in PEP-II to 30 reduce the fundamental impedance acting 20 Loop gain (dB) on the beam: direct and comb. 10 0 Direct loop is a proportional feedback −10 loop around the cavity. Closing the direct −20 −1500 −1000 −500 0 500 1000 1500 feedback loop reduces the effective Frequency (kHz) 3 impedance seen by the beam and lowers 10 the growth rates. Impedance (k Ω ) 2 10 To reduce the growth rates further we add the comb filter with narrow gain peaks at 1 synchrotron sidebands. 10 −1500 −1000 −500 0 500 1000 1500 Frequency (kHz) Expected growth rates shown here are Growth Damping Growth rate (ms −1 ) computed using a linear transfer function model of the RF feedback system. 0 10 According to the linear model the growth −2 rate reduction is two orders of magnitude, 10 −10 −5 0 5 10 Mode from 30 to 0.35 ms -1 Super-B Factory Workshop in Hawaii April 21, 2005
PEP-II low-level RF feedback: impedances and growth rates Two feedback loops are used in PEP-II to 30 reduce the fundamental impedance acting 20 Loop gain (dB) on the beam: direct and comb. 10 0 Direct loop is a proportional feedback −10 loop around the cavity. Closing the direct −20 −1500 −1000 −500 0 500 1000 1500 feedback loop reduces the effective Frequency (kHz) 3 impedance seen by the beam and lowers 10 the growth rates. Impedance (k Ω ) 2 10 To reduce the growth rates further we add the comb filter with narrow gain peaks at 1 synchrotron sidebands. 10 −1500 −1000 −500 0 500 1000 1500 Frequency (kHz) Expected growth rates shown here are Growth Damping Growth rate (ms −1 ) computed using a linear transfer function model of the RF feedback system. 0 10 According to the linear model the growth −2 rate reduction is two orders of magnitude, 10 −10 −5 0 5 10 Mode from 30 to 0.35 ms -1 Super-B Factory Workshop in Hawaii April 21, 2005
PEP-II low-level RF feedback: impedances and growth rates Two feedback loops are used in PEP-II to 30 reduce the fundamental impedance acting 20 Loop gain (dB) on the beam: direct and comb. 10 0 Direct loop is a proportional feedback −10 loop around the cavity. Closing the direct −20 −1500 −1000 −500 0 500 1000 1500 feedback loop reduces the effective Frequency (kHz) 3 impedance seen by the beam and lowers 10 the growth rates. Impedance (k Ω ) 2 10 To reduce the growth rates further we add the comb filter with narrow gain peaks at 1 synchrotron sidebands. 10 −1500 −1000 −500 0 500 1000 1500 Frequency (kHz) Expected growth rates shown here are Growth Damping Growth rate (ms −1 ) computed using a linear transfer function model of the RF feedback system. 0 10 According to the linear model the growth −2 rate reduction is two orders of magnitude, 10 −10 −5 0 5 10 Mode from 30 to 0.35 ms -1 Super-B Factory Workshop in Hawaii April 21, 2005
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