Quality factor Figure-of-merit for any oscillator Stored energy - - PowerPoint PPT Presentation

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Quality factor Figure-of-merit for any oscillator Stored energy - - PowerPoint PPT Presentation

Feb 19, 2024 38 likes •158 views

Quality factor Figure-of-merit for any oscillator Stored energy In the system Quality (or Q- ) factor Power dissipated per cycle P comes from imperfection in the system For copper (normal-conducting) cavities this

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SLIDE 1
  • P. Piot, PHYS 571 – Fall 2007

Quality factor

  • Figure-of-merit for any oscillator
  • P comes from “imperfection” in the system

– For copper (normal-conducting) cavities this “imperfection” is Ohmic loss Q=106-7 – For superconducting cavities (no Joule heating) these imperfections come from aperture use to couple the power to the cavity along with other needed instrument Q=1010 – Non-∞ Q → the cavity resonate for a finite time

Stored energy In the system Power dissipated per cycle Quality (or Q-) factor

ω τ Q ~

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SLIDE 2
  • P. Piot, PHYS 571 – Fall 2007

Quality factor

  • Q is also representative of the system bandwidth
  • Same as in RLC

circuits

ω ω ∆ = Q 1

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SLIDE 3
  • P. Piot, PHYS 571 – Fall 2007

Q for TM mode

The quality factor writes

  • r
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SLIDE 4
  • P. Piot, PHYS 571 – Fall 2007

Q for TE mode

⇒

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SLIDE 5
  • P. Piot, PHYS 571 – Fall 2007

Q for TE mode

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SLIDE 6
  • P. Piot, PHYS 571 – Fall 2007

Comparison with JDJ’s notations

  • Note on shunt impedance
  • Q-factor notations

(these Notes) (JDJ) (JDJ) (from these Notes)

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SLIDE 7
  • P. Piot, PHYS 571 – Fall 2007

Comparison with JDJ’s notations

  • The Geometric factor
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SLIDE 8
  • P. Piot, PHYS 571 – Fall 2007
  • Example of TE111 mode

Comparison with JDJ’s notations

Same as JDJ 8.97

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SLIDE 9
  • P. Piot, PHYS 571 – Fall 2007
  • Consider a volume V bounded by the surface S the force on the S

due to the e.m. field present in V is computed from the Maxwell’s stress tensor via

  • With and
  • Introducing the displacement
  • The work done by the e.m. field against the displacement is

Cavity perturbation & Slater’s theorem I

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SLIDE 10
  • P. Piot, PHYS 571 – Fall 2007
  • Note the quantity is a pressure (see your PHYS 563)

since at the conductor boundary we have

  • So
  • Time-averaging gives

Cavity perturbation & Slater’s theorem II

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SLIDE 11
  • P. Piot, PHYS 571 – Fall 2007
  • Consider the TM010 mode (“famous” mode in accelerators) at r=0
  • Compute the frequency shift associated to a small object at r=0, z

Cavity perturbation & Slater’s theorem III

By measuring δω/ω as the small

  • bject is moved

along z we can reconstruct the z- dependence of Ez

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SLIDE 12
  • P. Piot, PHYS 571 – Fall 2007
  • Example of real measurement

[Brown et al. PRSTAB 4 083501]

Slater’s theorem: example of application

Bead pull move z=0 mm Iris z=20 mm