Shunt impedance

To produce longitudinal Coulomb forces which add up to the (longitudinal) acceleration voltage ${\displaystyle \scriptstyle V_{\parallel }}$, an eigenmode of the resonator has to be excited, leading to power dissipation ${\displaystyle \scriptstyle P}$. The definition of the longitudinal effective shunt impedance, ${\displaystyle \scriptstyle R}$, then reads:^[2]

${\displaystyle R={\frac {|V_{\parallel }|^{2}}{P}}}$

with the longitudinal effective acceleration voltage ${\displaystyle \scriptstyle |V_{\parallel }|}$.

The time-independent shunt impedance, ${\displaystyle \scriptstyle R_{0}}$, with the time-independent acceleration voltage ${\displaystyle \scriptstyle V_{0}}$ is defined:^[2]

${\displaystyle R_{0}={\frac {V_{0}^{2}}{P}}.}$

One can use the quality factor ${\displaystyle \scriptstyle Q}$ to substitute ${\displaystyle \scriptstyle P}$ with an equivalent expression:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R = Q \frac{|V_\parallel|^2}{\omega W},}

where W is the maximum energy stored. Since the quality factor is the only quantity in the right equation term that depends on wall properties, the quantity ${\displaystyle \scriptstyle {\frac {R}{Q}}}$ is often used to design cavities, omitting material properties at first (see also cavity geometry factor).

When a particle is deflected in transverse direction, the definition of the shunt impedance can be used with substitution of the (longitudinal) acceleration voltage by the transverse effective acceleration voltage, taking into account transversal Coulomb and Lorentz forces.

${\displaystyle R_{\perp }={\frac {|V_{\perp }|^{2}}{P_{0}}}=Q{\frac {|V_{\perp }|^{2}}{\omega W}}}$

This does not necessarily imply a change in particle energy since a particle can also be deflected by magnetic fields (see Panofsky-Wenzel theorem).

Because the transverse deflection can be described with polar coordinates, one may define a deflection or polarization angle using the transverse acceleration voltage components. Polar coordinates are used because it is possible to add up voltage components like vectors, but not shunt impedances.