**Gain Ramping for Transverse Stochastic Cooling Systems in the
Debuncher**

Dave McGinnis

November 15, 2004

[Transverse Stochastic Cooling Equation]{.underline}

For transverse stochastic cooling, the cooling term is given as:

$\frac{d\left( a^{2} \right)_{\text{rms}}}{\text{dt}} = - \left( a^{2} \right)_{\text{rms}}2f_{o}K\sum_{m = - \infty}^{\infty}g_{m}$
(1)

$K = \frac{1}{\text{pc}q}\sqrt{\frac{Z_{\text{pu}}Z_{\text{kr}}}{2}}\frac{\sqrt{\beta_{\text{pu}}\beta_{\text{kr}}}}{h_{\text{pu}}}\text{ef}_{o}$
(2)

where f~o~ is the revolution frequency, pc is the momentum of the beam,
Z~pu~ and Z~kr~ is the pickup and kicker impedance, ~pu~ and ~kr~ are
the beta functions at the pickup and kicker, h~pu~ is the effective
height of the pickup, and g~m~ is the electronic gain at the m^th^
betatron sideband. The heating term due to other particles is:

$\frac{d\left( a^{2} \right)_{\text{rms}}}{\text{dt}} = \left( a^{2} \right)_{\text{rms}}N_{p}2f_{o}K^{2}\sum_{m = - \infty}^{\infty}{\frac{\left| g_{m} \right|^{2}}{|m|}\frac{f_{o}}{{\Delta f}_{o}}}$
(3)

where N~p~ is the number of particles and f~o~ is the spread in
revolution frequencies. The heating term due to amplifier noise is:

$\frac{d\left( a^{2} \right)_{\text{rms}}}{\text{dt}} = \frac{\left( a^{2} \right)_{\text{rms}}N_{p}}{S\left( \left( a^{2} \right)_{\text{rms}} \right)}2f_{o}K^{2}\sum_{m = - \infty}^{\infty}\left| g_{m} \right|^{2}$
(4)

where S((a^2^)~rms~) is the average signal to noise and is given as:

$S\left( \left( a^{2} \right)_{\text{rms}} \right) = \frac{\frac{\beta_{\text{pu}}\left( a^{2} \right)_{\text{rms}}}{h_{\text{pu}^{2}}}N_{p}Z_{\text{pu}}\left( \text{qf}_{o} \right)^{2}}{k_{B}T_{\text{pu}}f_{o}}$
(5)

where k~B~ is Boltzman's constant (1.36x10^-23^ Watts K^-1^Hz^-1^) and
T~pu~ is the effective noise temperature of the pickup and pickup
pre-amplifier. If the gain of the electronics is flat over a bandwidth
W, then the r.m.s. betatron amplitude is given as:

$\frac{d\left( a^{2} \right)_{\text{rms}}}{\text{dt}} = - 4W\left( a^{2} \right)_{\text{rms}}\left( \text{Kg}(t) - \left( \text{Kg}(t) \right)^{2}N_{p}\left( M(t) + \frac{1}{S\left( \left( a^{2} \right)_{\text{rms}} \right)} \right) \right)$
(6)

where M(t) is the mixing factor:

$M(t) = \frac{f_{o}}{W}\frac{f_{o}}{{\Delta f}_{o}(t)}\text{ln}\left( \frac{f_{\text{max}}}{f_{\text{min}}} \right)$
(7)

where f~max~ and f~min~ are the maximum and minimum frequencies of the
cooling system.

[Gain Ramping]{.underline}

The output power of the cooling system is given as:

$P(t) = \left( P_{\text{bo}}\left( \frac{P_{b}(t)}{P_{\text{bo}}} \right) + P_{n} \right)\left( \frac{\text{Kg}(t)}{\text{Kg}_{o}} \right)^{2}$
(8)

where P~bo~ is the initial beam power, P~n~ is the noise power, and g~o~
is the initial gain. The noise power can be written as the initial beam
power divided by the initial signal to noise:

$P_{n} = \frac{P_{\text{bo}}}{S_{o}}$ (9)

The total initial power is:

$P_{n} = \frac{P_{\text{bo}}}{S_{o}}$ (10)

The ratio of beam power to initial beam power is equal to the ratio of
the emittance to the initial emittance:

$\frac{P_{b}(t)}{P_{\text{bo}}} = \frac{\left( a(t)^{2} \right)_{\text{rms}}}{\left( a_{o}^{2} \right)_{\text{rms}}}$
(11)

Equation 8 becomes

$P(t) = \frac{P_{o}}{1 + \frac{1}{S_{o}}}\left( \frac{\left( a(t)^{2} \right)_{\text{rms}}}{\left( a_{o}^{2} \right)_{\text{rms}}} + \frac{1}{S_{o}} \right)\left( \frac{\text{Kg}(t)}{\text{Kg}_{o}} \right)^{2}$
(12)

The gain profile as a function of power is:

$\frac{\text{Kg}(t)}{\text{Kg}_{o}} = \sqrt{\frac{P(t)}{P_{o}}\frac{S_{o} + 1}{S_{o}\frac{\left( a(t)^{2} \right)_{\text{rms}}}{\left( a_{o}^{2} \right)_{\text{rms}}} + 1}}$
(13)

[Example]{.underline}

The gain ramping in the following example is set so that the total
cooling system power is kept constant throughout the entire cycle. The
parameters used in the example are shown in Table 1.

  ----------------------- ------------------ -----------------------------
  f~min~                  4.5                GHz

  f~max~                  7.5                GHz

  Production              15                 x10^-6^

  MI Beam                 5                  x10^12^

  f~o~                    590020             Hz

                          0.006              

  pc~o~                   8801               MeV

  pc~init~                80                 MeV

  pc~asym~                5                  MeV

  ~pc~                    1                  Sec

  ~95init~                20                 -mm-mrad

  ~init~                  1.3                Sec
  ----------------------- ------------------ -----------------------------

*Table 1.*

The revolution frequency spread for the mixing factor is computed from
the following:

$\frac{{\Delta f}_{o}(t)}{f_{o}} = \eta\frac{\left( \text{pc}_{\text{init}} - \text{pc}_{\text{asym}} \right)e^{- \frac{t}{\tau_{\text{pc}}}} + \text{pc}_{\text{asym}}}{\text{pc}_{o}}$
(14)

The parameter ~init~ is the initial cooling rate which is used to
determine the initial electronic gain:

$\frac{1}{\tau_{\text{init}}} = 4\text{WKg}_{o}$ (15)

Figure 1 shows the emittance as a function of time for an initial
average signal to noise of three with and without gain ramping. Using
the same average initial signal to noise ratio, Figure 2 shows the ratio
between the cooling to heating term with and without gain ramping. For
this example, the heating term does not contribute substantially to the
cooling rate. Figure 3 shows the emittance at the end of two seconds of
cooling as a function of the initial signal to noise ratio. If the
initial signal to noise ratio is three, then using gain ramping reduces
the emittance by 35%.

![](media/image1.wmf)

*Figure 1. Emittance as a function of time for an initial average signal
to noise of three with and without gain ramping.*

![](media/image2.wmf)

*Figure 2. The ratio between the cooling to heating term with and
without gain ramping.*

![](media/image3.wmf)

*Figure 3. The emittance at the end of two seconds of cooling as a
function of the initial signal to noise ratio*