Difference: Ctf3Sensitivity (14 vs. 15)

Revision 1511 Feb 2014 - FrankieCullinan

Line: 1 to 1

Cavity Sensitivity Estimates

Line: 62 to 62
Digitiser in 4.44 26.0 7.11 30.0

  • Position scale factor: 0.62 mm

Finite Decay Time

Because the signal decays with a finite decay time, the assumption that the root mean square voltage over one period is the peak voltage divided by $\sqrt{2}$ is no longer valid. The corresponding value can be found by

 \begin{equation*} \frac{V_{rms}(\tau)}{\hat{V}}=\int_0^T e^{-\frac{2t}{\tau}}\sin^2 (\omega t) dt = \frac{1}{2}\int_0^T e^{-\frac{2t}{\tau}}(1-\cos (2 \omega t) dt \end{equation*}
where ω is the angular frequency, τ is the decay time and T is the time of one period in time t. Evaluating the integral gives
 \begin{equation*} \frac{V_{rms}(\tau)}{\hat{V}}=\frac{\tau}{4T}\left(1-\frac{1}{1+\omega^2\tau^2}\right)\left(1-e^{-\frac{T}{\tau}}\right) \end{equation*}
This is one case where the signal is given by a sine function. In the opposing case, where a cosine function is used instead, the expression evaluates to
 \begin{displaymath} \frac{V_{rms}(\tau)}{\hat{V}}=\frac{\tau}{4}\left(1+\frac{1}{1+\omega^2\tau^2}\right)\left(1-e^{-\frac{T}{\tau}}\right)\, . \end{displaymath}
Another effect is the signal decaying before the first peak in the oscillation is seen. The peak time in the case of a

Latex rendering error!! dvi file was not created.

META FILEATTACHMENT attachment="possense.png" attr="h" comment="" date="1361106460" name="possense.png" path="possense.png" size="82817" stream="possense.png" user="Public.FrankieCullinan" version="2"
META FILEATTACHMENT attachment="refsense.png" attr="h" comment="" date="1361106346" name="refsense.png" path="refsense.png" size="86511" stream="refsense.png" user="Public.FrankieCullinan" version="1"
META FILEATTACHMENT attachment="latexd5e79cee3c33e10c50caf2c7c59c139b.png" attr="h" comment="" date="1392112892" name="latexd5e79cee3c33e10c50caf2c7c59c139b.png" user="pvap079" version="1"
META FILEATTACHMENT attachment="peak.pdf" attr="h" comment="" date="1392113079" name="peak.pdf" path="peak.pdf" size="14135" user="pvap079" version="1"
META FILEATTACHMENT attachment="power.pdf" attr="h" comment="" date="1392113079" name="power.pdf" path="power.pdf" size="21245" user="pvap079" version="1"
This site is powered by the TWiki collaboration platform Powered by PerlCopyright © 2008-2023 by the contributing authors. All material on this collaboration platform is the property of the contributing authors.
Ideas, requests, problems regarding RHUL Physics Department TWiki? Send feedback