Application of Non-Linear Time-Domain RF Simulations to Longitudinal Emittance Studies for the LHC

A non-lin­ear time-do­main sim­u­la­tion has been de­vel­oped that can de­ter­mine tech­ni­cal lim­i­ta­tions, ef­fects of non-lin­ear­i­ties and im­per­fec­tions, and im­pact of ad­di­tive noise on the in­ter­ac­tion of the beam with the Impedance Con­trol Radio Fre­quen­cy (RF) sys­tems [1]. We pre­sent a for­mal­ism for the ex­trac­tion of pa­ram­e­ters from the time-do­main sim­u­la­tion to de­ter­mine the sen­si­tiv­i­ty of the beam lon­gi­tu­di­nal emit­tance and di­lu­tion on the RF sys­tem char­ac­ter­is­tics. Pre­vi­ous stud­ies [2], [3] have es­ti­mat­ed the ef­fect of a noise source on the beam char­ac­ter­is­tics as­sum­ing an in­de­pen­dent per­tur­ba­tion source of the RF volt­age and a sim­pli­fied beam model with no cou­pling. We pre­sent the method­ol­o­gy for the time-do­main sim­u­la­tion study of the de­pen­dence of the ac­cel­er­at­ing volt­age noise spec­trum on the var­i­ous RF pa­ram­e­ters and the tech­ni­cal prop­er­ties (such as non-lin­ear­i­ties, ther­mal noise, fre­quen­cy re­sponse etc.) of the Low Level RF (LLRF) sys­tem com­po­nents. Fu­ture plans to ex­pand this for­mal­ism to cou­pled bunch stud­ies of lon­gi­tu­di­nal emit­tance growth in the LHC at nom­i­nal and up­grad­ed beam cur­rents are briefly sum­ma­rized.