By J. Corbett, A. Terebilo
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E. the entire orbit perturbation is due to an rf-frequency perturbation. ) approximation since betatron orbits tend to be oscillatory with the betatron period, and the rf-orbit is predominately DC. To compute the rf frequency shift required to correct the orbit we have ∂f rf = − f (Δf rf ) where Δfrf is the change in frequency used to generate the measured dispersion orbit, Δxrf. The residual betatron component, xβ = x - f Δxrf , is corrected with standard methods for orbit control. MATLAB Example – Projection orbit into dispersion vector >>edit rf_1 AUGMENTED RESPONSE MATRIX An alternative way to correct the rf orbit is to add column vector Δxrf to the response matrix prior to inversion.
Can the corrector magnets put the beam back on the original orbit? If the response matrix has sufficient rank ( the orbit perturbation lies in the column space of R) we can steer the beam through the BPMS - but what does the corrector pattern look like and what happens in-between the BPMS? Arguably, if we reduce the average corrector strength then the magnetic flux encircled by the beam, and consequently the beam energy, is reduced back toward the original value. In practice, this is indeed the case, a corrector pattern with a strong DC-component will compensate rf-frequency variations.
1 ⎢ 0 wn ⎥⎦ ⎢⎣ un ⋅ x ⎥⎦ ⎣ (W diagonal) ⎡ w1 −1u1 ⋅ x ⎤ ⎥ ⎢ θ =V ⋅⎢ | ⎥ ⎢ wn −1un ⋅ x ⎥ ⎦ ⎣ (multiply by inverse gains wi-1) −1 | ⎤ ⎡ w1 u1 ⋅ x ⎤ ⎡| ⎥ ⎢ θ = ⎢v1 ... v m ⎥ ⋅ ⎢ | ⎥ ⎢ ⎥ −1 | ⎥⎦ ⎢⎣ wn un ⋅ x ⎥⎦ ⎢⎣ | (expansion by eigenvectors) θ = ∑ ( wi −1ui ⋅ x )v i i The final equation expresses the corrector set θ as a linear combination of corrector eigenvectors vi. The coefficients of the expansion are projections of the orbit vector x into the orbit eigenvectors ui weighted by the inverse singular values wi-1.