Rise time of cascaded blocks

Hello everybody,
I am studying electronics at university, and, during the course of "electronic measurements" I encountered the following formula for the calculation of the the rise time of cascaded blocks:

tr_tot = sqrt ( tr_1 ^ 2 + tr_2 ^ 2 + ... + tr_n ^ 2 )

where tr_tot is the total rise time, and tr_i is the rise time of i-th block.
I have been looking for the theoretical explanation of this formula for a very long time in the internet (Wikipedia, specialized web-sites, etc.) without finding anything useful, and browsing many forums about electronics, I found out that other people have been looking for the same demonstration too.
However, Wikipedia explains that this formula derives from the "central limit theorem" without explaining why.
Recently I have find the following PDF file in the internet:

https://www.google.com/url?sa=t&rct...sg=AFQjCNE85SEQPoqE3JnfkRKQQ4Llohr3Ow&cad=rja

This documentation, published by Hewlett-Packard in 1949, explains that the formula is valid, in theory, only in case the frequency responses of the blocks is a Gaussian curve. If this is true, since the the total frequency response of cascaded blocks is the product of the frequency responses, the formula is derived. Please look at the file for further details. Considering that, the formula becomes a good approximation for blocks whose frequency response is not exactly a Gaussian.
I hope I was helpful for a lot of people that, like me, have been looking for the same explanation. I hope I didn't violate any protocol introducing this document. Let me know what you think.
Thank you very much for your attention.
Ciao

Let me know what you think.

Ciao.

Hi Francesco,

thanks for the interesting HP publication.

I have used this formula always to explain the rise time displayed by an oscilloscop in case the bandwidth of the y channel matters.
In this case, the internal rise time adds to the risetime to be measured - and it must be a geometrical summation as indicated by the formula.
However, as mentioned by you, the formula is 100% valid for lowpass systems exhibiting a gaussian characteristic only.
However, it also is a good approximation for critically damped systems consisting of a series of 1st order systems.
Thus, if the oscilloscop y-channel is approximated by a first-order system this formula can be used with advantage to correct measurements of
rise times influenced by internal oscilloscop rise times.

In case of 1st order systems the rise time is inversely proportional to the individual 3dB corner frequencies.
For a series combination of two blocks you, therfore, can write:

(1/wc)^2=(1/wc1)^2 + (1/wc2)^2.

This equation is the result of the following calculation:
*Multiply both transfer functions of 1st order
*Find the magnitude of the combined function
*Set the denominator equal to sqrt(2)
*Solve for wc
*For this purpose, the resulting 4th order equation (magnitude denumerator) is expanded as a series
using the approximation/simplification: sqrt(1+x)=1+0.5x
*This approximation results in a max. error of 6% (worst case wc1=wc2)
For all other cases the error is smaller.
*As a result, you get the formula for (1/wc)^2 as given above.
* Finally, you can replace 1/wc by t_rise.
___________________________________________________
Hope this helps to understand the formula.
Regards
LvW

---------- Post added at 10:40 ---------- Previous post was at 10:15 ----------

Sorry, my explanation above contains a small error:
Not the denumerator is expanded in form of a series but the square root expression sqrt(1+x).
This series is cut after the 2nd element resulting in the mentioned approximation (1+0.5x).

Thank you very much LvW.
I have been trying to solve this by myself using the Laplace transform for a very long time, without making it. Perhaps, I could not solve the problem because I have always been tempted to to calculate the rise time of the resulting transfer function in the time domain (by definition), without passing through the calculation of the cut-off frequency. I want to thank you very much for this tip. I still am not expert in electronics, but that I have always had a strong will to understand things.
However, since I am still interested in this subject, I want to ask you if you can suggest me some further material to read that I can find in the internet, or some books that I can borrow from the library of my university. This would be very useful for me.
Thank you very much, and I hope everyone will forgive me for my not perfect english.
Ciao.
Francesco

Hi Francesco,

I am sorry, but I am not able to give you any literature information on this subject. I don`t remember the source I have found (and used) many years ago to derive the formula under discussion. However, if you are interested (and if you don't succeed to derive the formula for wc by yourself) I can give you, of course, the whole calculation with details.
And - if I am allowed (as a german) to comment on this - your english is perfect enough to exchange questions and answers in this forum.
Regards
LvW