1 Tuesday Week 2 Lecture Jeff Eldred Wakefield Impedances 1 1 1 1 1

2 Wakefield Impedance 2 2 2 2 2

Wake Functions Decaying cosine-like Decaying sine-like From Panofsky Wenzel: 3 3 3 3

Impedance Analogy to EE Impedances as a force per current: Impedances as a Fourier decomposition of Wake functions: 4 4 4 4

Tabulated Impedances Many common sources of wakefield impedances – space-charge, resistive wall, RF cavities, BPMs, beampipe transitions, etc. – have been already been calculated. MAFIA, CST are able to perform simulations for more complicated scenarios. 5 5 5 5

Broad vs. Narrow Impedances 6 6 6 6

Impedance Compensation Impedance from one source can be compensated by impedances from another source. Some impedances are capacitive and others are reactive and consequently the imaginary part of the impedance can be canceled in the complex plane. Example: IMPEDANCE CALCULATION FOR FERRITE INSERTS Ferrite inserts to compensate space-charge driven impedances without introducing undesired new impedances. 7 7 7 7

Toy Model: Oscillator with Feedback The real part is a tune-shift, the imaginary part is an exponential growth/damping term. Different sources of impedance can actually be used to cancel each other out. 8 8 8 8

It’s More Complicated than that The wakefields drive the motion of the beam, but there is only a feedback loop if the motion of the particles in the beam adds up coherently. If the beam is focused nonlinearly then Landau damping will counteract the instability. Should be modeled as set of independently moving oscillators driven by a collective force. 9 9 9 9

Multipole Expansion 10 10 10 10

It’s More Complicated than that Next USPAS Chao and Stupakov are teaching a 2-week course on collective effects. Chao’s K.Y. Ng’s 11 11 11 11