Difference: Ctf3Sensitivity (15 vs. 16)
Revision 1611 Feb 2014 - FrankieCullinan
| Line: 1 to 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
Cavity Sensitivity Estimates | |||||||||
| Line: 66 to 66 | |||||||||
Finite Decay TimeBecause 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a finite decay time, the assumption that the root mean square voltage over one period is the peak voltage divided by is no longer valid. The corresponding value can be found by | |||||||||
| Added: | |||||||||
| > > | ![\[ \left(\frac{V_{rms}(\tau)}{\hat{V}}\right)^2=\frac{1}{T} \int_0^T e^{-\frac{2t}{\tau}}\sin^2 (2 \pi f t) dt = \frac{1}{2T}\int_0^T e^{-\frac{2t}{\tau}}(1-\cos (4 \pi f t)) dt \]](/twiki/pub/PP/JAI/Ctf3Sensitivity/latexfa5ca9771b0c8721bc3a056c31644848.png) | ||||||||
| %BEGINLATEX% | |||||||||
| Deleted: | |||||||||
| < < | \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*}
%ENDLATEX%
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
 Latex rendering error!! dvi file was not created. | ||||||||
| \begin{displaymath} | |||||||||
| Changed: | |||||||||
| < < | \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)\, . | ||||||||
| > > | \left(\frac{V_{rms}(\tau)}{\hat{V}}\right)^2=\frac{\tau}{4T}\left(1-\frac{1}{1+(2 \pi f \tau)^2}\right)\left(1-e^{-\frac{2 T}{\tau}}\right) | ||||||||
| \end{displaymath} %ENDLATEX% | |||||||||
| Added: | |||||||||
| > > | 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
![\[ \left(\frac{V_{rms}(\tau)}{\hat{V}}\right)^2=\frac{\tau}{4T}\left(1+\frac{1}{1+(2 \pi f \tau)^2}\right)\left(1-e^{-\frac{2T}{\tau}}\right)\, . \]](/twiki/pub/PP/JAI/Ctf3Sensitivity/latexd9c775860f8cf3f3cb4ac9983f1bb78f.png) ![\[ \frac{V_{rms}(\tau)}{\hat{V}}=\sqrt{\frac{\tau}{4T}\left(1-e^{-\frac{2T}{\tau}}\right)}\, . \]](/twiki/pub/PP/JAI/Ctf3Sensitivity/latexc2a28d54bc230c4df2ddf10c15af199d.png) | ||||||||
Another effect is the signal decaying before the first peak in the oscillation is seen. The peak time in the case of a
| |||||||||
| Line: 104 to 104 | |||||||||
| |||||||||
| Added: | |||||||||
| > > |
| ||||||||
View topic | History: r16 < r15 < r14 < r13 | More topic actions...
Copyright © 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