Accelerator Physics Topic I Acceleration

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## Accelerator Physics Topic I Acceleration

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**Accelerator PhysicsTopic IAcceleration**Joseph Bisognano Synchrotron Radiation Center University of Wisconsin J. J. Bisognano**Relativity**J. J. Bisognano**Maxwell’s Equations**J. J. Bisognano**Vector Identity Games**Poynting Vector Electromagnetic Energy J. J. Bisognano**Propagation in Conductors**J. J. Bisognano**Free Space Propagation**J. J. Bisognano**Conductive Propagation**J. J. Bisognano**Boundary Conditions**J. J. Bisognano**AC Resistance**J. J. Bisognano**Cylindrical Waveguides**• Assume a cylindrical system with axis z • For the electric field we have • And likewise for the magnetic field J. J. Bisognano**Solving for Etangential**J. J. Bisognano**Maxwell’s equations then imply (k=/c)**J. J. Bisognano**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**TE Rectangular Waveguide Mode**x a y b J. J. Bisognano**a TE mode Example**J. J. Bisognano**Circular Waveguide TEm,nModes**J. J. Bisognano**Circular Waveguide TEm,nModes**J. J. Bisognano**Circular Waveguide TMm,nModes**J. J. Bisognano**Circular Waveguide Modes**J. J. Bisognano**Cavities**d J. J. Bisognano**Cavity Perturbations**Now following C.C. Johnson, Field and Wave Dynamics J. J. Bisognano**Cavity Energy and Frequency**I ++++ E B - - - - -I Attracts Repels J. J. Bisognano**Energy Change of Wall Movement**J. J. Bisognano**Bead Pull**J. Byrd J. J. Bisognano**Lorentz Theorem**• Let and be two distinct solutions to Maxwell’s equations, but at the same frequency • Start with the expression J. J. Bisognano**Vector Arithmetic**J. J. Bisognano**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**Scattering Matrix**• Consider a multiport device • Discussion follows Altman S2 Sp S1 J. J. Bisognano**S-matrix**• Let apamplitude of incident electric field normalize so that ap2 = 2(incident power) and bp2 = 2(scattered power) J. J. Bisognano**Two-Port Junction**Port X Port Y a2 a1 b2 b1 J. J. Bisognano**Implication of Lorentz Theorem**J. J. Bisognano**Lorentz/cont.**• Lorentz theorem implies • or J. J. Bisognano**Unitarity of S-matrix**• Dissipated power P is given by • For a lossless junction and arbitrary this implies J. J. Bisognano**Symmetrical Two-Port Junction**J. J. Bisognano**Powering a Cavity**b1 b2 a1 a2 J. J. Bisognano**Power Flow**J. J. Bisognano**Power Flow/cont.**J. J. Bisognano**Optimization**• With no beam, best circumstance is ; I.e., no reflected power J. J. Bisognano**At Resonance**J. J. Bisognano**Shunt Impedance**• Consider a cavity with a longitudinal electric field along the particle trajectory • Following P. Wilson z2 z1 J. J. Bisognano**Shunt Impedance/cont**J. J. Bisognano**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**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**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**Beam Loading/cont.**Ve e V2 q V1 +V2 V1 J. J. Bisognano**Beam Loading/cont.**• With negligible loss • But particle loses • Since q is arbitrary, e =0 and J. J. Bisognano**Beam Loading/cont.**• Note: we have same constant (R/Q) determining both required power and charge-cavity coupling J. J. Bisognano**Beam Induced Voltage**• Consider a sequence of particles at J. J. Bisognano**Summary of Beam Loading**• References: Microwave Circuits (Altman); HE Electron Linacs (Wilson, 1981 Fermilab Summer School) J. J. Bisognano**Vector Addition of RF Voltages**Vb Vc Vg y j Vgr Vbr y q Vb J. J. Bisognano