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The `PILLBOX` command provides an analytical model for a cylindrical RF cavity. Both TM~mnp~ and `TE_mnp` modes for latexmath:[m \ge 0], latexmath:[n \ge 1] and latexmath:[p \ge 0] \(TM) and latexmath:[p \ge 1] (TE).
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.A block equation
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[latexmath]
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[latexmath]
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++++
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E_r = -\frac{p \pi}{L} \frac{R}{x_{mn}} E_0 J_m^\prime(k_{mn} r) \cos(m \varphi) \sin(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t)\\
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E_\varphi = -\frac{p \pi}{L} \frac{m R^2}{x_{mn}^2 r} E_0 J_m(k_{mn} r) \sin(m \varphi) \sin(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t)\\
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E_r = -\frac{p \pi}{L} \frac{R}{x_{mn}} E_0 J_m^\prime(k_{mn} r) \cos(m \varphi) \sin(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t)\\
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E_\varphi = -\frac{p \pi}{L} \frac{m R^2}{x_{mn}^2 r} E_0 J_m(k_{mn} r) \sin(m \varphi) \sin(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t)\\
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E_z = E_0 J_m(k_{mn} r) \cos(m \varphi) \cos(\frac{p\pi}{L} z)\exp(\dot{\iota} \omega t) \\
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B_r = -\dot{\iota} \omega \frac{m R^2}{x_{mn}^2 r c^2} E_0 J_m(k_{mn} r) \sin(m \varphi) \cos(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t) \\
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B_\varphi = -\dot{\iota} \omega \frac{R}{x_{mn} c^2} E_0 J_m^\prime (k_{mn} r) \cos(m \varphi) \cos(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t) \\
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B_z = 0
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++++
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where latexmath:[ L ] is the length and latexmath:[ R ] the radius of the pillbox, latexmath:[ J_m ] is the m-th Bessel function of the first kind, latexmath:[ x_{mn} ] is the n-th root of latexmath:[ J_m ] and latexmath:[ k_{mn} = \frac{x_{mn}}{R}].
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And for the TE~mnp~:
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.A block equation
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[latexmath]
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++++
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E_r = \dot{\iota} \omega \frac{m R^2}{x_{mn}^{\prime 2} R} B_0 J_m(k_{mn} r)\sin(m \varphi) \sin(\frac{p\pi}{L} z) \exp(\dot{\iota} \omega t) \\
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E_\varphi = \dot{\iota} \omega \frac{R}{x_{mn}^\prime} B_0 J^\prime_{m}(k_{mn} r)\cos(m \varphi) \sin(\frac{p\pi}{L} z) \exp(\dot{\iota} \omega t) \\
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E_z = 0 \\
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B_r = \frac{p\pi}{L}\frac{R}{x_{mn}^\prime} B_0 J_m^\prime(k_{mn} r)\cos(m \varphi) \cos(\frac{p\pi}{L} z) \exp(\dot{\iota} \omega t) \\
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B_\varphi = -\frac{p \pi}{L}\frac{m R^2}{x_{mn}^{\prime 2} r} B_0 J_m(k_{mn} r) \sin(m \varphi) \cos(\frac{p\pi}{L} z) \exp(\dot{\iota} \omega t) \\
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B_z = B_0 J_m(k_{mn} r) \cos(m \varphi) \sin(\frac{p\pi}{L}z) \exp(\dot{\iota} \omega t)
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++++
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where latexmath:[ L ] is the length and latexmath:[ R ] the radius of the pillbox, latexmath:[ J_m ] is the m-th Bessel function of the first kind, latexmath:[ x_{mn}^\prime ] is the n-th root of the first derivative of latexmath:[ J_m ] and latexmath:[ k_{mn} = \frac{x\
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_{mn}^\prime}{R}].
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[[sec.elements.travelingwave]]
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=== Traveling Wave Structure
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