the Accumulator ring the shunt impedance of a single RF cavity is 40 kΩ. There are 7 cavities, so the total shunt impedance is =280 kΩ. The total number of protons is Np Σ =1.5x1013, and the revolution time is =1.53x10-6 s. Therefore, = 1.5 Trev eNp / Trev= Σ A is the total beam current. A product = =425 kV is the total voltage, induced in Σ Σ×Σ the cavities for the maximum design parameters. The magnitude of is large, exceeding the voltage amplitude of the RF generator itself Σ (we have been considering 170 kV). This is why for design studies it is necessary to deal with a fractions of : =χ×Σ ,. An auxiliary scaling parameter , is a VBL Σ χ dimensionless beam loading factor, varying within [0, χmax]: (2) In numerical studies below we have been taking different from the interval [0, χ and were analyzing the beam dynamics: χmax=1] , correspond to ==425, 42, 21, and 4.2 kV, (3) VBL as it will be used in our calculations The goal is to find the upper limit of which still guarantees the quality bunching and this threshold will put limitations on the product (=VBL limiting the beam IDC×RΣ =), current and the shunt impedance of the cavities in each specific case. Ideally, for =1, if the bunching is appropriate, we approach the design parameters: = =425 kV. Σ Σ×Σ The Fourier coefficients after normalization: are plotted in the Figs. 5a, 5b for the beam loading with kV. Fig.5a Normalized Fourier coefficients (left), (right), for the si(χ,n) ci(χ,n) i=1,…,16 beam loading factor (), depending on the number of turns: turns (solid n=2500 χ=0.06 red), 5000 (dashed green), 10000 (dotted brown), 20000 (dash-dot blue). One can see the steady growth of all 4th multiple harmonics, during the formation of 4 bunches in the Accumulator. And Fig. 5b shows (n) (blue), (n) (red) for s4 c4
the Accumulator ring the shunt impedance of a single RF cavity is 40 kΩ. There are 7 cavities, so the total shunt impedance is Σ =280 kΩ. The total number of protons is Np =1.5x1013, and the revolution time is Trev =1.53x10-6 s. Therefore, Σ = eNp / Trev= 1.5 A is the total beam current. A product Σ = Σ×Σ =425 kV is the total voltage, induced in the cavities for the maximum design parameters. The magnitude of Σ is large, exceeding the voltage amplitude of the RF generator itself (we have been considering 170 kV). This is why for design studies it is necessary to deal with a fractions of Σ : VBL =χ×Σ ,. An auxiliary scaling parameter χ , is a dimensionless beam loading factor, varying within [0, χmax]: (2) In numerical studies below we have been taking different χ from the interval [0, χmax=1] and were analyzing the beam dynamics: , correspond to VBL ==425, 42, 21, and 4.2 kV, (3) as it will be used in our calculations The goal is to find the upper limit of which still guarantees the quality bunching and this threshold will put limitations on the product IDC×RΣ (=VBL =), limiting the beam current and the shunt impedance of the cavities in each specific case. Ideally, for =1, if the bunching is appropriate, we approach the design parameters: Σ = Σ×Σ =425 kV. The Fourier coefficients after normalization: are plotted in the Figs. 5a, 5b for the beam loading with kV. Fig.5a Normalized Fourier coefficients si(χ,n) (left), ci(χ,n) (right), i=1,…,16 for the beam loading factor χ=0.06 (), depending on the number of turns: n=2500 turns (solid red), 5000 (dashed green), 10000 (dotted brown), 20000 (dash-dot blue). One can see the steady growth of all 4th multiple harmonics, during the formation of 4 bunches in the Accumulator. And Fig. 5b shows s4 (n) (blue), c4 (n) (red) for
BL_mu2e_preprint_draft.docx