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Re: Back of the envelope Dynamic range
Hi folks,
There are a couple of misconceptions I think lurking around here...
First a small factor: naively treating the ion drift in the FDC as a
cylindrical chamber, applying standard formulas for instance from a
paper of Radeka, I get the signal charge is
Q(t)=Q0 * ln(1+t/t0) / ( 2*ln(rc/ra) ).
Here Q0 is the _total_ signal charge to time infinity;
rc is the "cathode radius", I'll use 5mm;
ra is the effective anode radius, I'll use 13um (it must be a little
larger than the wire, the mean radius of ion production in the avalanche
is what you put here);
t0 is the time scale parameter, t0=ra/(2*mu*V), I'll use mu=1.62e-4 and
V=1650, then t0=1.88ns;
With this Q(10ns) = 15% of Q0. 20% is too high a figure I think.
Secondly, it is all a little too simplistic to pretend that the preamp
measures just a charge and that it is exactly the charge delivered in
10ns. Both the signal and the preamp/shaper/ADC system have a frequency
response to consider here, and the overall response to detector current
is not simply an integration (g(s)=1/s) response over all the relevant
frequencies. We are not measuring precisely the charge or the current
but something in between. Back-of-the-envelope "charge" discussions
about the signal size may be useful to set the rough scale, but we need
someday to converge on a real specification, and that can only come from
a calculation or simulation that includes the less trivial
time/frequency response of the system.
Thirdly, it is obvious both a priori and from a quick look at signals
from the prototype, that not all the electrons arrive at the anode wire
simultaneously. Of course details of the arrival time distribution
depend on the track parameters, but I _guess_ it's probably fair to
assume a fluctuating fraction maybe 10% to 50% of the signal charge
actually arrives at the preamp in "the largest" 10 ns interval. I think
all the FDC/CDC signal size calculations I have seen written so far have
been ignoring this...
Fourth: a capacitance match on the preamp (Cdet=Camp) is in reference
to the _open_-loop amplifier; the closed-loop preamplifier is supposed
to have a much lower input impedance than that. _All_ of the charge is
supposed to go into the preamp (if the far side of the detector is not
having a termination resistor, of course; and of course this is also
ignoring the non-infinite coupling capacitance and charge sharing
between preamp and detector due to that). [I'm also not sure that this
ASIC is subject to this capacitive match rule, it does not directly
apply for such short peaking times, maybe?] In any case, this means a
factor of 1/2 for "capacitance match" is _not_ present and should be
dropped out of Fernando's slide.
[Ok Elton also already addressed this one by now...]: Fifth, I don't
know much about it but the figure of 100 (or 90) e-ion pairs produced by
the track, is the mean, isn't it? And the distribution of it (Landau
distribution?) has a significant tail. The implication being, we can't
just use the mean value to set the signal range of the readout
electronics or else it's going to saturate on a lot of events. Some
factor needs to be included, to account for the ratio between point
where remaining tail is ignorable and the mean. Unless I goofed the
math, I find a factor 2.2 should be included, for 5% of events are more
than a factor 2.2 greater than the mean. [Elton says 3.0.]
Gerard
p.s. Elton in your summary, you don't mention the CDC track inclination
(path length) contribution. Isn't that important? What is the
distribution of track lengths in the CDC? (I think Ryan produced a plot
of this for us once... I don't know if it is still accurate today.)
Elton Smith wrote:
> HI Fernando,
>
> Thanks for the plot. We had ignored your Cdet/Camp factor of 0.5. Your
> gain for the FDCs should be updated to 8^10^4. We agree that the dynamic
> range is fine for the FDCs.
>
> Curtis should comment on 1) operating gain of the CDCs (2x10^4?), and 2)
> the required dynamic range. My guesstimate is factor of 10 for dE/dx and a
> factor of 3 for Landau fluctuations, giving a total of 30. Linearity of
> this entire range is probably not required, as even saturation would
> identify slow protons. Clearly the region of linearity is most important
> for the FDC cathodes which matches the dynamic range.
>
> Thanks, Elton.
>
>
>
>
> Elton Smith
> Jefferson Lab MS 12H5
> 12000 Jefferson Ave
> Suite # 16
> Newport News, VA 23606
> elton@jlab.org
> (757) 269-7625
>
> On Wed, 14 Feb 2007, Fernando J. Barbosa wrote:
>
>
>>Hi Elton,
>>
>>I made a slide which I am attaching. Please check the numbers as I show
>>what a MIP in the chamber corresponds to the charge into the preamp.
>>
>>Regards,
>>Fernando
>>
>>Elton Smith wrote:
>>
>>>Hi Fernando,
>>>
>>>Yesterday during your presentation you showed the dynamic range of the
>>>preamp to saturate at 400 fC. We are a little confused about the range
>>>required. I know that there was quite a bit of discussion at the time the
>>>range was decided, but simple estimates give higher numbers and we need to
>>>know what we are missing.
>>>
>>>Fernando was correct in that the induced signal on the wire on the wire
>>>coming from the electron signal is only about 20% of the avalanch charge
>>>(in the first say 10 ns). Therefore, we obtain the following for a minimum
>>>ionizing track:
>>>
>>>Simon says that the gain of the FDC is about 0.8x10^5.
>>>
>>>assume: 100 ions, 20% of Qtot, Gain=10^5, e=1.6x10^-19.
>>>Signal charge (anode) = 0.2 x 100 x 10^5 x 1.6x10^-19 ~ 300 fC
>>>Signal charge (cathode) ~ 300 fC x 1/2 x 1/4 ~ 40 fC.
>>>
>>>My preliminary (rough) estimate concludes that this should work for the
>>>FDC cathodes. The FDC anodes under the nominal design do not use the
>>>FADCs, but would likely saturate if they do.
>>>
>>>For the CDC, the anode signal will also be used for dE/dx, so a factor of
>>>at least 30 is needed in dynamic range. These would clearly saturate at a
>>>Gain of 10^5, but we might want to run them at lower gain. At G=10^4
>>>gain, and 30 dynamic range, we get a maximum signal of 900 fC, also
>>>saturating the preamp.
>>>
>>>Comments? What am I doing wrong?
>>>
>>>Thanks, Elton.
>>>
>>>
>>>
>>>
>>>
>>>
>>>Elton Smith
>>>Jefferson Lab MS 12H5
>>>12000 Jefferson Ave
>>>Suite # 16
>>>Newport News, VA 23606
>>>elton@jlab.org
>>>(757) 269-7625
>>>
>>
>