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Accelerator Physics Topic I Acceleration

Accelerator Physics Topic I Acceleration. Joseph Bisognano Synchrotron Radiation Center University of Wisconsin. Relativity. Maxwell’s Equations. Vector Identity Games. Poynting Vector. Electromagnetic Energy. Propagation in Conductors. Free Space Propagation. Conductive Propagation.

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Accelerator Physics Topic I Acceleration

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  1. Accelerator PhysicsTopic IAcceleration Joseph Bisognano Synchrotron Radiation Center University of Wisconsin J. J. Bisognano

  2. Relativity J. J. Bisognano

  3. Maxwell’s Equations J. J. Bisognano

  4. Vector Identity Games Poynting Vector Electromagnetic Energy J. J. Bisognano

  5. Propagation in Conductors J. J. Bisognano

  6. Free Space Propagation J. J. Bisognano

  7. Conductive Propagation J. J. Bisognano

  8. Boundary Conditions J. J. Bisognano

  9. AC Resistance J. J. Bisognano

  10. Cylindrical Waveguides • Assume a cylindrical system with axis z • For the electric field we have • And likewise for the magnetic field J. J. Bisognano

  11. Solving for Etangential J. J. Bisognano

  12. Maxwell’s equations then imply (k=/c) J. J. Bisognano

  13. All this implies that E0zandB0z tell it all with their equations • For simple waveguides, there are separate solutions with one or other zero (TM or TE) • For complicated geometries (periodic structures, dielectric boundaries), can be hybrid modes J. J. Bisognano

  14. TE Rectangular Waveguide Mode x a y b J. J. Bisognano

  15. a TE mode Example J. J. Bisognano

  16. Circular Waveguide TEm,nModes J. J. Bisognano

  17. Circular Waveguide TEm,nModes J. J. Bisognano

  18. Circular Waveguide TMm,nModes J. J. Bisognano

  19. Circular Waveguide Modes J. J. Bisognano

  20. Cavities d J. J. Bisognano

  21. Cavity Perturbations Now following C.C. Johnson, Field and Wave Dynamics J. J. Bisognano

  22. Cavity Energy and Frequency I ++++ E B - - - - -I Attracts Repels J. J. Bisognano

  23. Energy Change of Wall Movement J. J. Bisognano

  24. Bead Pull J. Byrd J. J. Bisognano

  25. Lorentz Theorem • Let and be two distinct solutions to Maxwell’s equations, but at the same frequency • Start with the expression J. J. Bisognano

  26. Vector Arithmetic J. J. Bisognano

  27. Using curl relations for non-tensor m, e one can show that expression is zero • So, in particular, for waveguide junctions with an isotropic medium we have S2 S3 S1 J. J. Bisognano

  28. Scattering Matrix • Consider a multiport device • Discussion follows Altman S2 Sp S1 J. J. Bisognano

  29. S-matrix • Let apamplitude of incident electric field normalize so that ap2 = 2(incident power) and bp2 = 2(scattered power) J. J. Bisognano

  30. Two-Port Junction Port X Port Y a2 a1 b2 b1 J. J. Bisognano

  31. Implication of Lorentz Theorem J. J. Bisognano

  32. Lorentz/cont. • Lorentz theorem implies • or J. J. Bisognano

  33. Unitarity of S-matrix • Dissipated power P is given by • For a lossless junction and arbitrary this implies J. J. Bisognano

  34. Symmetrical Two-Port Junction J. J. Bisognano

  35. Powering a Cavity b1 b2 a1 a2  J. J. Bisognano

  36. Power Flow J. J. Bisognano

  37. Power Flow/cont. J. J. Bisognano

  38. Optimization • With no beam, best circumstance is ; I.e., no reflected power J. J. Bisognano

  39. At Resonance J. J. Bisognano

  40. Shunt Impedance • Consider a cavity with a longitudinal electric field along the particle trajectory • Following P. Wilson z2 z1 J. J. Bisognano

  41. Shunt Impedance/cont J. J. Bisognano

  42. Shunt Impedance/cont. • Define • where P is the power dissipated in the wall (the term) • From the analysis of the coupling “b” • where is the generator power J. J. Bisognano

  43. Beam Loading • When a point charge is accelerated by a cavity, the loss of cavity field energy can be described by a charge induced field partially canceling the existing field • By superposition, when a point charge crosses an empty cavity, a beam induced voltage appears • To fully describe acceleration, we need to include this voltage • Consider a cavity with an excitation V and a stored energy • What is ? J. J. Bisognano

  44. Beam Loading/cont. • Let a charge pass through the cavity inducing and experiencing on itself. • Assume a relative phase • Let charge be bend around for another pass after a phase delay of  J. J. Bisognano

  45. Beam Loading/cont. Ve e V2 q V1 +V2 V1 J. J. Bisognano

  46. Beam Loading/cont. • With negligible loss • But particle loses • Since q is arbitrary, e =0 and J. J. Bisognano

  47. Beam Loading/cont. • Note: we have same constant (R/Q) determining both required power and charge-cavity coupling J. J. Bisognano

  48. Beam Induced Voltage • Consider a sequence of particles at J. J. Bisognano

  49. Summary of Beam Loading • References: Microwave Circuits (Altman); HE Electron Linacs (Wilson, 1981 Fermilab Summer School) J. J. Bisognano

  50. Vector Addition of RF Voltages Vb Vc Vg y j Vgr Vbr y q Vb J. J. Bisognano

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