Computing normalized shunt impedances $R/Q$

There are several definitions for a normalized shunt impedance floating around. We take this one:

$$R/Q = \frac{V V^*}{2 \omega W}$$ (4.2)

where

• $R/Q$ is the normalized shunt impedance,
• $V$ is the complex voltage that would be seen by a test charge traversing the cavity at a speed of $\beta c_0$,
• $\omega$ is the circular frequency of the mode,
• $W$ is the total stored energy in the cavity (both electric and magnetic energy).

If one evaluates the voltage seen by the test particle, one arrives at the result

$$V = \int\limits_{z=z_1}^{z=z_2} E_z(x,y,z) e^\frac{{\rm j}\omega z}{\beta c_0}\; dz$$ (4.3)

for a particle that travels in positive z-direction from $z=z_1$ to $z=z_2$.