More information on impedance is available on the CERN impedance page

https://impedance.web.cern.ch/impedance/

The wikipedia page is starting to develop

https://en.wikipedia.org/wiki/Impedance_%28accelerator_physics%29

The impedance is a complex function given by the Fourier transform of the Wakefield:

(1)
\begin{align} Z(\omega) = \frac{1}{2\pi} \int W(s) e^{i \omega s} ds \end{align}

The broadband resonance impedance is given by

(2)
\begin{align} Z_{||}(\omega) = R_s \frac{1-i Q(\frac{\omega_r}{\omega}-\frac{\omega}{\omega_r})}{1+Q^2\left(\frac{\omega_r}{\omega}-\frac{\omega}{\omega_r}\right)^2} \end{align}

For the low frequency limit, we get

(3)
\begin{align} Z_{||}(\omega) = R_s \frac{\omega-i Q\omega_r}{1+Q^2\left(\frac{\omega_r}{\omega}\right)^2} \end{align}

or

(4)
\begin{align} Z_n = -i\frac{\omega_0}{\omega_r}\frac{R}{Q} \end{align}

Standard BBR for ESRF is $f_{res}=30\ GHz$, $R_s = 42\ kOhm$, $Q=1$.

Boussard criterion says instability when

(5)
\begin{align} \sigma_z > \frac{c}{f_s} \end{align}

One sometimes wants to know the kick factor and the loss factor.

For the BBR, the kick factor is given by

(6)
\begin{align} \kappa_{BBR}=\frac{c R}{2\sqrt{\pi}\sigma_s Q} \end{align}

The loss factor for a BBR is given by

(7)
\begin{align} \kappa_{L,BBR} = \frac{\omega_r R_s}{4\sqrt{\pi} Q^2 (k_r \sigma_s)^3} \end{align}

with

(8)
\begin{align} k_r = \frac{\omega_r}{c} \end{align}

A formula for the threshold of the microwave instability is given by

(9)
\begin{align} I_{th} = \frac{9.4 E_0 \nu_s^2}{e\alpha_c Z_n}\left(\frac{\omega_0 \sigma_s}{c}\right)^3 \end{align}

In Cai's paper, "Linear theory of microwave instability in electron storage rings" (2011), he defines a normalized current:

(10)
\begin{align} I_n = \frac{r_e N_b}{2\pi \nu_s \gamma \sigma_\delta} \end{align}

For the broadband resonator, the unitless parameters are given by

(11)
\begin{align} \xi = I_n R \omega_R \end{align}

and

(12)
\begin{align} \nu_r = \frac{\sigma_z \omega_r}{c} \end{align}

We want to compare this with the parameter $\Delta$ defined by Limborg and others.

(13)
\begin{align} \Delta = -\frac{2\pi I_b Z_n}{V_{rf} h\cos\phi \left(\frac{\alpha\sigma_\delta}{\nu_s}\right)^3} \end{align}

We can also write $\Delta$ as

(14)
\begin{align} \Delta=\frac{\alpha e I_b}{E_0 \nu_s^2}\left(\frac{c}{\omega_{0}\sigma_{z0}}\right)^3 Z_n \end{align}

The bunch current may be connected to the number of electrons in the bunch via

(15)
\begin{align} I_b = \frac{N_b e c_l}{c_R} \end{align}

with $c_l$ the speed of light and $c_R$ the ring circumference.