# Control of Transverse Multibunch Instabilities in the First Stage of the VLHC

#### Valeri Lebedev

### Fermilab

> VLHC mini-workshop
>
> SLAC, March 21-23, 2001.

##### Content

1.  Machine Parameters

2.  Multibunch Transverse Instability

3.  Feedback System for Suppression of the Transverse Multibunch
    Instability

4.  Emittance Growth Suppression

5.  Conclusions

VLHC Parameters

+----------------------------------+-----+-------------+-------------+
|                                  |     | **Stage 1** | **Stage 2** |
+----------------------------------+-----+-------------+-------------+
| Beam energy                      | *E~ | 20 TeV      | 87.5 TeV    |
|                                  | p~* |             |             |
+----------------------------------+-----+-------------+-------------+
| Luminosity                       | ##  | 10^34^      | 2⋅10^34^    |
|                                  | ### | cm^-2^s^-1^ | cm^-2^s^-1^ |
|                                  | # L |             |             |
+----------------------------------+-----+-------------+-------------+
| Magnetic field                   | *B* | 1.96 T      | 10 T        |
|                                  | ~0~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Injection energy                 | *E  | 0.9 TeV     | 10 TeV      |
|                                  | ~in |             |             |
|                                  | j~* |             |             |
+----------------------------------+-----+-------------+-------------+
| Bunch spacing                    | *τ~ | 18.9 ns     |             |
|                                  | b~* |             |             |
+----------------------------------+-----+-------------+-------------+
| Circumference                    | *C* | 232 km      |             |
|                                  | =2* |             |             |
|                                  | πR* |             |             |
+----------------------------------+-----+-------------+-------------+
| Revolution frequency             | *f* | 1294 Hz     |             |
|                                  | ~0~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Number of bunches                | *N~ | 36943       |             |
|                                  | b~* |             |             |
+----------------------------------+-----+-------------+-------------+
| RF frequency                     | f~  | 477.9 MHz   |             |
|                                  | RF~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Betatron tunes                   | *ν* | \~214       |             |
|                                  | ~⊥~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Momentum compaction              | *α* | 2.1⋅10^-5^  |             |
+----------------------------------+-----+-------------+-------------+
| Beta-function at IP              | *β* | 30 cm       | 50 cm       |
|                                  | ^∗^ |             |             |
+----------------------------------+-----+-------------+-------------+
| Head-on beam-beam tune shift per | *ξ* | 1.8⋅10^-2^  | 1.8⋅10^-2^  |
| IP                               |     |             |             |
+----------------------------------+-----+-------------+-------------+
| Number of particles per bunch    | ##  | 2.5⋅10^10^  |             |
|                                  | ### |             |             |
|                                  | # N |             |             |
+----------------------------------+-----+-------------+-------------+
| Beam current                     | *I~ | 0.190 A     | 0.175 A     |
|                                  | b~* |             |             |
+----------------------------------+-----+-------------+-------------+
| RF voltage per turn,             | *V* | 50/50 MeV   | 50/50 MeV   |
| top/injection energy             | ~0~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Synchrotron frequency,           | *f~ | 2.32/10.9   | 1.1/3.28 Hz |
| top/injection energy             | s~* | Hz          |             |
+----------------------------------+-----+-------------+-------------+
| Rms momentum spread,             | σ   | 1.5/        | 0.5         |
| top/injection energy             | ~p~ | 14.9⋅10^-4^ | /2.4⋅10^-4^ |
+----------------------------------+-----+-------------+-------------+
| Rms bunch length, top/injection  | σ   | 6.6/14.2 cm | 4.5/7.8 cm  |
| energy                           | ~s~ |             |             |
+----------------------------------+-----+-------------+-------------+

## 2. Transverse Impedance for Vacuum Chamber with Thin Wall

## Impedance calculation for round vacuum chamber 

$$\begin{array}{r}
k = \frac{1 - i}{\delta}\ ,\ \delta = \frac{c}{\sqrt{2\text{πσω}}}\ ,\  \\
Z_{0} = \frac{4\pi}{c} \approx \text{377}\ \Omega\ \text{.}
\end{array}$$

Solutions for the vector potential in different regions

$$A_{z}(r,\theta,t) = \begin{bmatrix}
A_{I} \\
A_{\text{II}} \\
A_{\text{III}}
\end{bmatrix} = \begin{bmatrix}
\begin{array}{r}
\frac{2I_{0}x_{0}}{\text{cr}} + C_{1}r \\

\end{array} \\
\begin{array}{r}
C_{2}e^{k(r - a)} + C_{3}e^{- k(r - a)} \\

\end{array} \\
\frac{C_{4}}{r}
\end{bmatrix}\ e^{i\omega t}\text{cos}\theta$$

Matching solutions at boundaries:
$\frac{\text{dA}}{d\theta}|_{r} = \frac{\text{dA}}{d\theta}|_{r + 0};\ \frac{\text{dA}}{\text{dr}}|_{r} = \frac{\text{dA}}{\text{dr}}|_{r + 0}$
yields four linear equations for coefficients *C~i~* . The solution is:

######## $$\begin{array}{r}
C_{1} = \frac{2I_{0}x_{0}}{\text{ca}^{2}}\left\lbrack 2\frac{e^{\text{kd}}\left( \text{bk} + 1 \right) + e^{- \text{kd}}\left( \text{bk} - 1 \right)}{e^{\text{kd}}\left( \text{ak} + 1 \right)\left( \text{bk} + 1 \right) - e^{- \text{kd}}\left( \text{ak} - 1 \right)\left( \text{bk} - 1 \right)} - 1 \right\rbrack\ , \\
C_{2} = \frac{4I_{0}x_{0}}{\text{ca}}\frac{e^{- \text{kd}}\left( \text{bk} - 1 \right)}{e^{\text{kd}}\left( \text{ak} + 1 \right)\left( \text{bk} + 1 \right) - e^{- \text{kd}}\left( \text{ak} - 1 \right)\left( \text{bk} - 1 \right)}\ , \\
C_{3} = \frac{4I_{0}x_{0}}{\text{ca}}\frac{e^{\text{kd}}\left( \text{bk} + 1 \right)}{e^{\text{kd}}\left( \text{ak} + 1 \right)\left( \text{bk} + 1 \right) - e^{- \text{kd}}\left( \text{ak} - 1 \right)\left( \text{bk} - 1 \right)}\ , \\
C_{4} = \frac{8I_{0}x_{0}}{\text{ca}}\frac{b^{2}k}{e^{\text{kd}}\left( \text{ak} + 1 \right)\left( \text{bk} + 1 \right) - e^{- \text{kd}}\left( \text{ak} - 1 \right)\left( \text{bk} - 1 \right)}\ \text{.}\ 
\end{array}$$

Taking into account contribution from electric field of the beam we
obtain an expression for the vacuum chamber transverse impedance (case
of positive frequency):

$Z_{} = - i\left( \frac{{\beta C}_{1}}{I_{0}x_{0}} + \frac{2}{\text{ca}^{2}\beta} \right) = - i\frac{Z_{0}}{2{\pi a}^{2}\beta}\left\lbrack 2\frac{\beta^{2}e^{\text{kd}}\left( 1 + \text{kb} \right) - e^{- \text{kd}}\left( 1 - \text{kb} \right)}{e^{\text{kd}}\left( 1 + \text{ka} \right)\left( 1 + \text{kb} \right) - e^{- \text{kd}}\left( 1 - \text{ka} \right)\left( 1 - \text{kb} \right)} + 1 - \beta^{2} \right\rbrack$
(1)

There are following asymptotes for the transverse impedance (*ω* \> 0):

$$Z_{} = Z_{0}\begin{array}{r}
\left\{ \frac{c(1 - i)}{2{\pi a}^{3}\sqrt{2\text{πσ}_{R}\omega}}\ ,\ \delta \leq d \middle| \right.\ \left\{ \frac{c^{2}}{4\pi^{2}\sigma_{R}{\omega a}^{3}d} - i\left( \text{.}\text{.}\text{.} \right)\ ,\ \sqrt{\text{ad}} \geq \delta \geq d \middle| \right.\ 
\end{array}$$

-   There is no simple expression for imaginary part of the impedance in
    the case $\sqrt{\text{ad}} \geq \delta \geq d$ and Eq. (1) has to be
    used

-   To simplify the solution the expression for the vector potential
    inside conductor was chosen for the flat geometry (instead of Bessel
    functions of imaginary argument) and therefore the result is not
    quite accurate if the wall is sufficiently thick.

    -   In particular the equation yields incorrect value for imaginary
        part of the impedance at very small frequencies,
        $\delta \geq \sqrt{\text{ad}}$

        -   The following correction term is applied to fix the problem

> $${\Delta Z}_{} = - i\frac{Z_{0}}{2{\pi a}^{2}}\frac{d}{(a + d)\left( 1 - k^{4}d^{2}a^{2} \right)}$$

-   Note that although this term corrects asymptotic behavior of the
    impedance at low frequencies it does not change the impedance at
    frequencies of interest of VLHC.

![](media/image3.wmf)![](media/image4.wmf)

> *d* = 0.1 mm *d* = 1 mm

Real (red) and imaginary (brown) parts of the transverse impedance and
asymptotes for the real part of the impedance as function of frequency
for round vacuum chamber: Al, 300 K, ρ = 2.74 μΩ⋅cm, *a* = 9 mm.

-   Skin-layer thickness is 3.3 mm at *f*~0~/2=646 Hz.

    -   Peak is achieved when $\delta^{2} \approx \text{ad}$ and its
        value does not depend on vacuum chamber thickness

**Transverse Impedance of Elliptic Vacuum Chamber**

## Impedance of flat vacuum chamber is about half of the round vacuum chamber impedance. For an estimate we will introduce the effective radius of the vacuum chamber: 

## $$a_{\text{eff}} = \left( \frac{2a^{3}b^{3}}{a^{3} + b^{3}} \right)^{\frac{1}{3}}$$

## so that we would get correct result for cases $a = b$, $a\text{>>}b$and $a\text{<<}b$.

## For impedance estimates we will substitute *a~eff~* instead of *a* into Eq.(1) 

####### Vacuum Chamber Conductivity

-   Vacuum chamber cooling significantly reduces transverse impedances

##  

## Detuning Wake

-   For round vacuum chamber the beam excites dipole field behind him.
    That corresponds to normal wake field *W*(*s*).

-   For non-round vacuum chamber there are quadrupole field and higher
    multipoles. Contribution corresponding to quadrupole field is
    described by detuning wake *D*(*s*):

$$\begin{array}{r}
\int_{}^{}{F_{x}ds^{'} = e^{2}\left( x_{0}W_{x}(s) + \text{xD}(s) \right)} \\
\int_{}^{}{F_{y}ds^{'} = e^{2}\left( y_{0}W_{y}(s) - \text{yD}(s) \right)}
\end{array}$$

-   Detuning wake changes betatron tunes for tail particles and can
    cause non-linear resonances.

    -   For round vacuum chamber

    -   *D*(*s*) = 0 .

    -   For two parallel plates

-   $$W_{x}(s) = W_{x}(s) = - D_{x}(s) \approx \frac{R}{a^{3}}\sqrt{\frac{c}{\sigma_{R}s}}$$

![](media/image11.wmf)

Attenuation of the beam magnetic field by the elliptic aluminum vacuum
chamber at 300 K,

*a* = 9 mm, *b* = 14 mm; red line - *d*=1 mm, blue line - *d*=2 mm

-   There is only about 30% attenuation for thickness of 1 mm at the
    first unstable betatron sideband

    -   That implies that the beam sees everything around vacuum chamber
        and can acquire additional contribution to the impedance from
        the surroundings if appropriate measures are not taken.

-   2 mm vacuum chamber attenuates the beam field by more than 2 times

    -   That is better but still requires careful attitude to what can
        be located in near vicinity of vacuum chamber

## Multibunch Transverse Instability due to Finite Wall Resistivity 

######### Estimate of tune shift and instability increment due to vacuum chamber impedance

$$W_{}(s) = \frac{i}{2\pi}\int_{- \infty}^{\infty}{{d\omega Z}_{}(\omega)e^{\frac{- i\omega s}{v}}},\ Z_{}(\omega) = - \frac{i}{v}\int_{0}^{\infty}{\text{dsW}_{}(s)e^{\frac{i\omega s}{v}}}$$

Let the beam position for mode *n* of continuous beam be as
$x(t,\theta) = x_{0}e^{i\left( (q - n)\omega_{o}t - n\theta \right)}$

Then, the force acting on the reference particle is

$$F_{0}(t) = \frac{\text{eI}_{b}}{v}\int_{0}^{s}{W(s)x_{0}e^{- i\left( (q - n)\omega_{0}\frac{s}{v} \right)}\text{ds}} = \text{ieI}_{b}Z_{}\left( - (q - n)\omega_{0} \right)x_{0}$$

$$\Rightarrow \ {\Delta\nu}_{n} = - i\frac{\text{eI}_{b}R^{2}}{2\ Pv\nu\ }Z_{}\left( - (q - n)\omega_{0} \right)\ $$

# The exact result for the betatron tune shift of *n*-th mode for the bunched beam

> $\begin{array}{r}
> {\Delta\nu}_{n} = - i\frac{\text{eI}_{b}R^{2}}{2\ Pv\nu\ }\sum_{k = - \infty}^{\infty}{Z_{}\left( - \omega_{\text{nk}} \right)}\ , \\
> \omega_{\text{nk}} = \omega_{0}\left( \lbrack q\rbrack + n + \text{Nk} \right)\ 
> \end{array}$ (2)

# \[q\] -- the fractional part of the betatron tune *ν*

> N -- the number of bunches,
> $n \in \left\lbrack \frac{- N}{2},\ (N - 1\frac{)}{2} \right\rbrack$
>
> $Z_{}(\omega)$ -- the transverse impedance per unit length averaged
> over the ring

# The Dimensionless Increment 

> $\frac{\lambda_{n}}{f_{0}} \equiv 2\pi\text{Im}(\Delta\nu) = - \frac{\pi\text{eI}_{b}R^{2}}{\ Pv\nu\ }\text{Re}\left( \sum_{k = - \infty}^{\infty}{Z_{}\left( - \omega_{\text{nk}} \right)} \right)$
> (3)

-   The definition of the transverse impedance yields that impedance at
    a negative frequency is related to the impedance at positive
    frequency through its complex conjugated value

$Z_{}( - \omega) = - Z_{{}^{}}(\omega)$

or

$$\begin{array}{r}
\text{Re}\left( Z_{}( - \omega) \right) = - \text{Re}\left( Z_{}(\omega) \right) \\
\text{Im}\left( Z_{}( - \omega) \right) = \text{Im}\left( Z_{}(\omega) \right)
\end{array}$$

-   At low frequencies impedance is completely determined by the wall
    resistivity.

    -   For *ω *\> 0 real part of the impedance is positive ⇒ coherent
        motion is damped

        -   $$\omega_{n} = \omega_{0}\left( \lbrack q\rbrack + n \right)\ ,\ n = 0\text{.}\text{.}\text{.}(N - 1\frac{)}{2}$$

    -   For *ω *\< 0 real part of the impedance is negative ⇒ coherent
        motion is unstable

        -   $$\left| \omega_{n} \right| = \omega_{0}\left( n - \lbrack q\rbrack \right)\ ,\ n = 1\text{.}\text{.}\text{.}\frac{N}{2}$$

> ⇒ If the impedance grows with frequency decrease preferable fractional
> tune is in the range \[0 -- 0.5\]

**Asymptotics for the dimensionless increment**

$$\frac{\lambda_{n}}{f_{0}} = - \frac{\pi\text{eI}_{b}R^{2}}{\ Pv\nu\ }Z_{0}\text{sign}\left( \omega_{n} \right)\begin{array}{r}
\left\{ \frac{c}{2{\pi a}^{3}\sqrt{2\text{πσ}_{R}\left| \omega_{n} \right|}}\ ,\ \delta \leq d \middle| \right.\ \left\{ \frac{c^{2}}{4\pi^{2}\sigma_{R}\left| \omega_{n} \right|a^{3}d}\ ,\ \sqrt{\text{ad}} \geq \delta \geq d \middle| \right.\ 
\end{array}$$

![](media/image12.wmf) ![](media/image13.wmf)

Dependence of the dimensionless decrement (left) and the betatron tune
shift (right) on the mode frequency for elliptic aluminum vacuum chamber
at 300 K, *a* = 9 mm, *b* = 14 mm;

red line - *d*=1 mm, blue line - *d*=2 mm; brown dashed line - *a* = 12
mm, *b* = 20 mm, *d*=2 mm.

-   9 mm vacuum chamber yields instability growth time about one
    revolution

    -   Increase of wall thickness from 1 mm to 2 mm decreases

        -   the growth time by about 1.5 times

        -   the coherent tune shift by about 2 times to *∆ν *\~ 0.1

-   To decrease decrement to about 2 revolutions one needs to increase
    vacuum chamber size to about 12×20 mm

    -   That also decreases the coherent tune shift to *∆ν *\~ 0.03 (30
        deg per turn)

## Coherent and incoherent tune shifts

-   The interaction of the beam with round vacuum chamber

    -   causes instabilities and changes frequencies of coherent beam
        motion

    -   but does not change incoherent particle frequencies and cannot
        cause non-linear resonance

-   In the case of elliptic vacuum chamber detuning wake is not zero.
    That means that the interaction of the beam with vacuum chamber
    creates additional focusing fields which changes incoherent particle
    tunes

    -   In the case of flat vacuum chamber and uniformly filled ring the
        frequency of all particles is shifted by the same amount:
        ${\Delta\nu}_{\text{ic}0} = - \frac{\pi^{2}}{\text{24}}\frac{\text{eI}_{b}R^{2}}{\ \text{mc}^{3}\text{γβ}^{3}a^{2}\nu\ }$

*∆ν~ic~*~0~= −0.205 for half-gap *a*=9 mm. It is equal to half of the
coherent tune shift at zero frequency. *∆ν~ic~*~0~ can be compensated by
betatron frequency adjustments proportional to the beam current

-   If the ring is partially filled then there is tune variations
    proportional to "AC" components in the beam current

    -   For flat vacuum chamber the wake and the detuning wake are equal
        and therefore tune variations can be computed using results
        obtained for coherent betatron tune shift

> $${\Delta\nu}_{\text{ic}_{n}} = \text{Re}(\Delta\nu) = - \frac{\text{eI}_{{b\omega}_{n}}R^{2}}{2\ Pv\nu\ }\text{Im}\left( \sum_{k = - \infty}^{\infty}{Z_{}\left( \omega_{n} \right)} \right)\ ,\ \omega_{n} = {n\omega}_{0}$$

-   To compensate tune variations along the beam the modulation of
    lattice betatron tune on revolution frequency is required

```{=html}
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```
-   Interaction of the beam DC current with iron of the dipoles causes
    additional incoherent tune shift

> $${\Delta\nu}_{\text{icB}} = - \frac{\pi^{2}}{\text{12}}\frac{\text{eI}_{b}R^{2}}{\ \text{mc}^{3}\text{γβ}\ a_{d^{2}}\nu\ }\eta$$
>
> *a~d~* -- is the dipole half-gap
>
> *η* -- fraction of the orbit with dipoles
>
> That tune shift is almost two times higher than the incoherent tune
> shift due to vacuum chamber ${\Delta\nu}_{\text{ic}0}$(*∆ν~icB~*=
> −0.332 for *a~d~*=10 mm)

-   If the ring is partially filled then there are tune variations
    proportional to "AC" components in the beam current. Their amplitude
    will depend on the thickness of vacuum chamber and dipole
    laminations.

## 4. Feedback System for Suppression of the Transverse Multibunch Instability

## Standard system

$$\begin{array}{r}
V_{n} = V_{0}\text{exp}\left( i\omega\frac{T}{N}n \right) \\
V_{n + N} = \begin{bmatrix}
\text{cos}(\mu_{0}) & \text{sin}(\mu_{0}) \\
 - \text{sin}(\mu_{0}) & \text{cos}(\mu_{0})
\end{bmatrix}\begin{bmatrix}
1 - k(\omega) & 0 \\
0 & 1 - k(\omega)
\end{bmatrix}V_{n} \\
k(\omega) = \sum_{j = 0}^{\infty}{g\left( \frac{T}{N}j \right)\text{exp}\left( - \frac{i\omega T}{N}j \right)}
\end{array}$$

> *N* -- number of bunches
>
> *T* -- revolution period
>
> *g*(*t*) -- response function

######## Perturbation theory solution

$$\begin{array}{r}
\Lambda_{n} \equiv \text{exp}\left( {i\omega}_{n}T \right) = \left( 1 - k({\overset{\sim}{\omega}}_{n}) \right)\text{exp}\left( {i\mu}_{0} \right) \\
\ {\overset{\sim}{\omega}}_{n}T = \mu_{0} + 2\pi n
\end{array}$$

[Damping decrement]{.underline}:
$\lambda_{n} = - \text{ln}\left( \left| \Lambda_{n} \right| \right)/T$

[Integrating type narrow band system]{.underline}

![](media/image16.wmf)

![](media/image17.wmf)

-   There is no principal limits to reach damping faster than the
    revolution time

-   Narrow band system can be stable in the entire frequency range

-   Technical problems

    -   Signal transfer from BPMs to kickers

        -   Forward transfer

            -   Expensive tunnel for signal transfer

        -   One turn delay

            -   Large betatron phase shift after one turn

            -   Large delay ⇒ digital system

        -   Practical system needs subtraction of orbit offset

            -   Slow feedback to nullify BPMs

            -   Or notch filter at revolution frequency which will
                reduce maximum achievable gain

    -   Limits of the system gain

        -   Final accuracy of electronics

        -   Emittance growth due to BPM noise

    -   Kicker voltage

        -   5 m, ±10 kV kicker suppresses 2.5 mm injection oscillations
            at 1 turn

## System with delayed response

[Single system with delayed response]{.underline}

$$\begin{array}{r}
V_{n} = V_{0}\text{exp}\left( i\omega\frac{T}{N}n \right) \\
V_{n + N} = \begin{bmatrix}
\text{cos}(\mu_{0}) & \text{sin}(\mu_{0}) \\
 - \text{sin}(\mu_{0}) & \text{cos}(\mu_{0})
\end{bmatrix}V_{n} - k(\omega)\begin{bmatrix}
0 & 0 \\
0 & 1
\end{bmatrix}V_{n + N} \\
k(\omega) = \sum_{j = 0}^{\infty}{g\left( \frac{T}{N}j \right)\text{exp}\left( - \frac{i\omega T}{N}\left( j + j_{d} \right) \right)}
\end{array}$$

> *N* -- number of bunches
>
> *T* -- revolution period
>
> *g*(*t*) -- response function
>
> *j~d~* -- number of skipped bunches

######## Perturbation theory solution

$$\begin{array}{r}
\Lambda_{n} \equiv \text{exp}(i\omega T) = \frac{1 + k(\omega_{n})/2}{1 + k(\omega_{n})}\text{cos}\left( \mu_{0} \right) \pm \sqrt{\left( \frac{1 + k(\omega_{n})/2}{1 + k(\omega_{n})} \right)^{2}\text{cos}^{2}\left( \mu_{0} \right) - \frac{1}{1 + k(\omega_{n})}} \\
\ \omega_{n}T = \mu_{0} + 2\pi n
\end{array}$$

[Integrating type system]{.underline}
$g(t) = g_{0}\text{exp}\left( - \frac{t}{\tau} \right)\ ,\ k(\omega) = \frac{g_{0}}{1 - \text{exp}\left( - \frac{T}{N}\left( \frac{1}{\tau} + i\omega \right) \right)}$

![](media/image22.wmf)*g*~0~=2⋅10^-4^, *τ*/*T*=.025, μ~0~=2π⋅0.27,
N=10^5^, *j~d~* =50 -- 50 skipped bunches (10^5^/(4⋅500))

Optimization strategy

-   Suppress resistive wall instability at low frequencies

-   Achieve minimum increments at higher frequencies

[Double system with delayed response]{.underline}

$$\begin{array}{r}
M = \begin{bmatrix}
\text{cos}(\mu_{0}) & \text{sin}(\mu_{0}) \\
 - \text{sin}(\mu_{0}) & \text{cos}(\mu_{0})
\end{bmatrix}\ ,\ M = M_{1}M_{2}\ , \\
l = 1,2 \\
\ k_{l}(\omega) = \sum_{j = 0}^{\infty}{g_{l}\left( \frac{T}{N}j \right)\text{exp}\left( - \frac{i\omega T}{N}\left( j + j_{d} \right) \right)} \\
\ k_{l} = \begin{bmatrix}
1 & 0 \\
0 & 1 + k_{l}(\omega)
\end{bmatrix}\ ,\  \\
\text{MV}_{n} = \left( k_{1} + M_{2}k_{2}M_{2^{- 1}}k_{1} \right)V_{n + N}
\end{array}$$

> *N* -- number of bunches
>
> *T* -- revolution period
>
> *g*(*t*) -- response function
>
> *j~d~* -- number of skipped bunches

$$\begin{array}{r}
\Lambda_{n^{2}}\left\lbrack 1 + k_{1}(\omega_{n}) + k_{2}(\omega_{n}) + k_{1}(\omega_{n})k_{2}(\omega_{n}) \right\rbrack - \\
\ 2\Lambda_{n}\left\lbrack \text{cos}\left( \mu_{0} \right)\left( 1 + \frac{k_{1}(\omega_{n}) + k_{2}(\omega_{n})}{2} \right) + \text{cos}\left( \mu_{1} \right)\text{cos}\left( \mu_{2} \right)\frac{k_{1}(\omega_{n})k_{2}(\omega_{n})}{2} \right\rbrack + 1 = 0
\end{array}$$

######## Comparison of single (*μ*~0~, *g*~0~) and double systems (*μ*~1~=*μ*~2~=μ~0~/2, *g*~1~=*g*~2~=*g*~0~)

######## ![](media/image25.wmf)

######## Dependence of absolute values of eigen-numbers on system gain, *g*, for different betatron tunes. 

######## ![](media/image26.wmf)

######## Damping decrement Residual instability increment

> N=10^5^, *j~d~* =50; Single system: *g*~0~=2⋅10^-4^,
> *τ*/*T*=2.5⋅10^-2^, μ~0~=2π⋅0.27,
>
> Double system: *g*~0~=8⋅10^-8^, *τ*/*T*=2⋅10^-2^, μ~0~=2π⋅0.46.

######## Conclusions for a system with delayed response for VLHC

-   System allows to reduce a requirement for the damping decrement of
    the bunch-bunch dumping system by an order of magnitude

-   Installation of multiple systems does not allow to increase the
    total damping due to anti-damping at high frequency.

-   Decreasing the number of skipped bunches one can improve performance
    but it increases noise effects due to smaller beta-functions

## 5. Emittance Growth Suppression

Emittance growth due to noises[^1]

$\frac{d\varepsilon}{\text{dt}} = {\pi f}_{0^{2}}\xi^{2}\sum_{n = - \frac{N}{2} + 1}^{\frac{N}{2}}{\left( \frac{3\text{.}3}{\left( 2\frac{\lambda_{n}}{f_{0}} \right)^{2} + \xi^{2}} + \text{67} \right)\left\langle \beta_{2}S_{\text{noise}}\left( \omega_{n} \right) \right\rangle}\ ,\ \omega_{n} = \omega_{0}\left( \left\lbrack \nu_{} \right\rbrack + n \right)$
,

Spectral density of transverse kicks,
$\overline{\theta_{{}^{2}}} = \int_{- \infty}^{\infty}{S_{\text{noise}}\left( \omega_{n} \right)d\omega}$,
for the feedback system

> is determined by spectral density of BPM noise $x_{\omega^{2}}$,
> system gain and beta-function at BPM, *β*~1~ :
>
> $\left\langle \beta_{2}S_{\text{noise}}\left( \omega_{n} \right) \right\rangle = \frac{x_{\omega^{2}}}{\beta_{1}}k^{2}\left( \omega_{n} \right)$
> .
>
> *β*~1~ -- beta-function at BPM
>
> *β*~2~ -- beta-function at kicker

[Theoretical limit for the standard narrow band system]{.underline}

-   Effective BPM noise of the feedback system is determined by the
    thermal noise of amplifiers is

$$\overline{x_{\omega^{2}}} = 2a^{2}\frac{\text{kT}\rho}{{\pi I}_{b^{2}}Z_{\text{BPM}^{2}}}$$

> Factor of 2 appeared because the system consists of two independent
> systems shifted by 90 deg in betatron phase* *
>
> *ρ* = 50 Ω, T=300 K, *I~b~*=0.19 A, *Z~BPM~*=12 Ω   ⇒
>
> $\overline{x_{\omega^{2}}}$=10^-20^ cm^2^ s,
> $\sqrt{\overline{x^{2}}}$= 0.02 μm (∆f=25 MHz)
>
> ⇒
> $\frac{\varepsilon_{0}}{\frac{d\varepsilon}{\text{dt}}} \approx$`<!-- -->`{=html}1.3⋅10^5^
> days - at the injection

-   Required BPM accuracy

> Emittance growth time \> 100 hour ⇒
>
> bunch position measured with $\sqrt{\overline{x^{2}}}$= 3 μm (180
> times of theoretical limit)

## Conclusions

-   Multi-bunch instabilities can be suppressed

-   To suppress the resistive wall instability

    -   Increase vacuum chamber thickness to 2 mm and vertical size to
        2×12 mm will be extremely profitable

    -   Two feedback systems required

        -   Low frequency system with high gain, narrow band and delayed
            response to damp low frequencies

            -   Unstable at high frequencies

        -   High frequency system to restore stability at high
            frequencies

            -   wide band, moderate gain and one turn delay

                -   Unstable at low frequencies

[^1]: ^^ Emittance growth due to noise and its suppression with the
    feedback system in large hadron colliders, V. Lebedev, *et.al.*
    V.V.Parkhomchuk, V.D.Shiltsev, G.V.Stupakov; SSCL-Preprint-188,
    Dallas, March 1993 and Particle Accelerators, 1994, v.44, pp.147-164